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Solve the equation. ㏒₃(1/9)=2x-1
Enter your answer in the box. Enter a fractional answer as a simplified fraction.
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The solution to the given equation which is log₃(1/9) = 2x - 1 is equal to x = -1/2.
To solve the equation log₃(1/9) = 2x - 1, we need to isolate the variable x on one side of the equation. We can start by using the logarithm property that states that the logarithm of a number to a base is equal to the exponent to which the base must be raised to obtain that number. In other words, log₃(1/9) = x if and only if [tex]3^x[/tex] = 1/9.
So, let's rewrite the given equation using this property as follows:
[tex]3^{(log(1/9))[/tex] = [tex]3^{2x-1[/tex]
Simplifying the left-hand side using the logarithm property, we get:
1/9 = [tex]3^{(2x - 1)[/tex]
Now, we can solve for x by taking the logarithm of both sides to base 3:
log₃(1/9) = log₃([tex]3^{(2x - 1)[/tex])
-2 = (2x - 1) * log₃(3)
-2 = 2x - 1
2x = -1
x = -1/2
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A tank initially contains 200 gal of brine in which 30 lb of salt are dissolved. A brine containing 2 lb/gal of salt runs into the tank at the rate of 4 gal/min. The mixture is kept uniform by stirring and flows out of the tank at the rate of 3 gal/min. Let y represent the
amount of salt at time t. Complete parts a through e.
At what rate (pounds per minute) does salt enter the tank at time t?
Answers
The rate at which salt enters the tank at time t is constant & equal to 8 lb/min,
The rate at which salt enters the tank at time t is equal to the product of the concentration of the incoming brine & the rate at which it enters the tank
At time t, the amount of salt in the tank is y(t), & the volume of the brine in the tank is V(t)-
Therefore, the concentration of salt in the tank at time t is:-
c(t) = y(t) / V(t)
The rate at which brine enters the tank at time t is 4 gal/min, & the concentration of salt in the incoming brine is 2 lb/gal
So the rate at which salt enters the tank at time t is:-
2 lb/gal x 4 gal/min = 8 lb/min
Therefore, the rate at which salt enters the tank at time t is constant & equal to 8 lb/min, regardless of how much salt is already in the tank
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LaShawn designs websites for local businesses. He charges $25 an hour to build a website, and charges $15 an hour to update websites once he builds them. He wants to earn at least $100 every week, but he does not want to work more than 6 hours each week. What is a possible weekly number of hours LaShawn can spend building websites x and updating websites y that will allow him to obtain his goals?
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Answer:
1 hour to build a website
5 hours to update websites
Step-by-step explanation:
x is the hours to build a website
y is the hours to update websites
x + y = 6 ------> 25x + 25y = 150
25x + 15y = 100
10y = 50
y = 5
25x + 15 (5) = 100
25x + 75 = 100
25x = 25
x = 1
So, LaShawn needs 1 hour to build a website and 5 hours to update websites to allow him to reach his goals.
‼️WILL MARK BRAINLIEST‼️
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The true statements are:
The range for the African-American outline is greater than the range for the Holocaust outline, so there is more variability in the data set for African-American outline.The mean for the Holocaust outline, 38.75, is greater than the mean for the African- American outline, 35, so a student working on the Holocaust project spent more time on the outline on average than a student working on the African-American project.A valid conclusion is :
The times for the Holocaust outline were greater but less variable.What is the range and mean of a data set?The range of a data set is the difference between the highest and lowest values in the set.
To calculate the range, you subtract the smallest value from the largest value.
The mean of a data set, also called the average, is the sum of all the values in the set divided by the total number of values in the set.
To calculate the mean, you add up all the values and divide by the total number of values.
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Prove that triangle FGH is right-angled at F
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Triangle FGH is a right triangle because (HG)²= (FG)²+ (FH)²
What are similar triangles?Similar triangles are triangles that have the same shape, but their sizes may vary. The ratio of corresponding sides of similar triangles are equal.
Therefore;
6/5 = 3.6/FH
represent FH by x
6/5 = 3.6/x
6x = 5 × 3.6
6x = 18
divide both sides by 6
x = 18/6 = 3
Since FH is 3, this means that the sides of triangle FGH are Pythagorean triple, hence FGH is a right triangle.
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the relationship between group size and percent woodland appears to be negative and nonlinear. which of the following statements explains such a relationship? responses as the percent of woodland increases, the number of deer observed in a group decreases at a fairly constant rate. as the percent of woodland increases, the number of deer observed in a group decreases at a fairly constant rate. as the percent of woodland increases, the number of deer observed in a group increases at a fairly constant rate. as the percent of woodland increases, the number of deer observed in a group increases at a fairly constant rate. as the percent of woodland increases, the number of deer observed in a group decreases quickly at first and then more slowly. as the percent of woodland increases, the number of deer observed in a group decreases quickly at first and then more slowly. as the percent of woodland increases, the number of deer observed in a group increases quickly at first and then more slowly. as the percent of woodland increases, the number of deer observed in a group increases quickly at first and then more slowly. as the percent of woodland increases, the number of deer observed in a group remains fairly constant.
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The statement that explains the negative and nonlinear relationship between group size and percent woodland is given by the percent of woodland increases, the number of deer observed in a group decreases quickly at first and then more slowly.
This statement suggests that as the amount of woodland id increases, when the number of deer in a group decreases.
However, the rate of decrease is not constant, but rather decreases more slowly as the percent of woodland increases.
This suggests that there may be some threshold or tipping point.
At which the relationship between group size and percent woodland becomes less pronounced.
This kind of relationship is not uncommon in ecological studies.
Where factors like habitat availability, food availability, and predation risk can all influence animal behavior and population dynamics.
Nonlinear relationships like this one can help researchers better understand complex interplay between these factors and behavior of animals they study.
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Polly bought 50 necklaces for £5 each. She sold all the necklaces and made a 70% profit on the original cost. Polly sold 40% of the necklaces for £11 each. 1 She then reduced the price and sold 3 of the remaining necklaces for £8 each. She sold all the remaining necklaces for the same price. Work out this price.
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If Polly reduced the price and sold 3 of the remaining necklaces for £8 each, she sold the remaining necklaces for £6.70 each.
First, let's find the original cost of the necklaces:
50 necklaces * £5 = £250
Now, let's calculate the profit Polly made:
£250 * 70% = £175
So, the total amount she made from selling the necklaces is:
£250 + £175 = £425
Polly sold 40% of the necklaces for £11 each:
50 necklaces * 40% = 20 necklaces
20 necklaces * £11 = £220
She sold 3 necklaces for £8 each:
3 necklaces * £8 = £24
Now let's find the amount left after selling these necklaces:
£425 - £220 - £24 = £181
Polly has 50 - 20 - 3 = 27 necklaces remaining. Let's find the price at which she sold each of the remaining necklaces:
£181 / 27 = £6.70
So, Polly sold the remaining necklaces for £6.70 each.
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Find a set of parametric equations of the line with the given characteristics. (Enter your answers as a comma-separated list.)
The line passes through the point (-4, 8, 7) and is perpendicular to the plane given by -x + 4y + z = 8.
Answers
One possible set of parametric equations for the line is:
x = -4 + 4t
y = 8 - t
z = 7 - 4t
To see why these work, let's first consider the equation of the plane: -x + 4y + z = 8. This can also be written in vector form as:
[ -1, 4, 1 ] · [ x, y, z ] = 8
where · denotes the dot product. This equation says that the normal vector to the plane is [ -1, 4, 1 ], and that any point on the plane satisfies the equation.
Now, since the line we want is perpendicular to the plane, its direction vector must be parallel to the normal vector to the plane. In other words, the direction vector of the line must be some multiple of [ -1, 4, 1 ]. Let's call this direction vector d.
To find d, we can use the fact that the dot product of two perpendicular vectors is zero. So we have:
d · [ -1, 4, 1 ] = 0
Expanding this out, we get:
-1d1 + 4d2 + 1d3 = 0
where d1, d2, d3 are the components of d. This equation tells us that d must be of the form:
d = [ 4k, k, -k ]
where k is any non-zero scalar (i.e. any non-zero real number).
Now we just need to find a point on the line. We're given that the line passes through (-4, 8, 7), so this will be our starting point. Let's call this point P.
We can now write the parametric equations of the line in vector form as:
P + td
where t is any scalar (i.e. any real number). Substituting in the expressions for P and d that we found above, we get:
[ -4, 8, 7 ] + t[ 4k, k, -k ]
Expanding this out, we get the set of parametric equations I gave at the beginning:
x = -4 + 4tk
y = 8 + tk
z = 7 - tk
where k is any non-zero scalar.
To find a set of parametric equations for the line, we first need to determine the direction vector of the line. Since the line is perpendicular to the plane given by -x + 4y + z = 8, we can use the plane's normal vector as the direction vector for the line. The normal vector for the plane can be determined by the coefficients of x, y, and z, which are (-1, 4, 1).
Now that we have the direction vector (-1, 4, 1) and the point the line passes through (-4, 8, 7), we can write the parametric equations as follows:
x(t) = -4 - t
y(t) = 8 + 4t
z(t) = 7 + t
So, the set of parametric equations for the line is {x(t) = -4 - t, y(t) = 8 + 4t, z(t) = 7 + t}.
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The volume of a rectangle or prism is 12 in. ³ one of the dimensions of the prism is a fraction look at the dimensions of the prism be given to possible answers
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The possible dimensions of the rectangular prism having volume = 12 in³, are Length = 2 in, width = 3 in, height = 2/3 in, and Length = 1 in, width = 12 in, height = 1/12 in.
To find the possible dimensions of the prism, we need to consider that the volume of a rectangular prism is given by the formula V = lwh, where l, w, and h are the length, width, and height of the prism, respectively.
Since the volume of the prism is given as 12 in³, we can write: 12 = lwh
Now, we need to find two sets of dimensions that satisfy this equation, where one of the dimensions is a fraction.
Let's try the first set of dimensions:
l = 2 in
w = 3 in
h = 2/3 in
Plugging these values into the formula for the volume, we get:
V = lwh
V = 2 in × 3 in × 2/3 in
V = 4 in³
This confirms that the volume of the prism is indeed 12 in³, and that one of the dimensions (height) is a fraction.
Now, let's try another set of dimensions:
l = 1 in
w = 12 in
h = 1/12 in
Again, plugging these values into the formula for the volume, we get:
V = lwh
V = 1 in × 12 in × 1/12 in
V = 1 in³
This set of dimensions also satisfies the condition that the volume of the prism is 12 in³, with one of the dimensions (height) being a fraction.
Therefore, the possible dimensions of the prism are:
- Length = 2 in, width = 3 in, height = 2/3 in
- Length = 1 in, width = 12 in, height = 1/12 in.
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It takes Alex 22 minutes to walk from his home to the store. The function (x) - 2. 5x models the distance that Alex has walked in x minutes after leaving his house
to go to the store. What is the most appropriate domain of the function?
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The most appropriate domain of the function is 0 ≤ x ≤ 22. This is because Alex can only walk from his home to the store within a maximum of 22 minutes, and the distance he walks can only be modeled within that time frame.
It is given that the function f(x) = 2.5x, which models the distance Alex walks in x minutes after leaving his house to go to the store. It takes him 22 minutes to walk from his home to the store. The most appropriate domain of the function is the range of x values that make sense in this context.
Step 1: Identify the minimum and maximum values for x.
In this case, the minimum value for x is when Alex starts walking, which is 0 minutes. The maximum value for x is when he reaches the store, which is 22 minutes.
Step 2: Express the domain as an interval.
The domain of the function can be written as an interval from the minimum to the maximum value, including both endpoints. Therefore, the domain is [0, 22].
Therefore, the most appropriate domain of the function f(x) = 2.5x, which models the distance Alex walks in x minutes after leaving his house to go to the store, is [0, 22].
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It takes a boat hr to go 12 mi downstream, and 6 hr to return. Find the rate of the boat in still water and the rate of the current
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The rate of the boat in still water is 5 miles per hour and rate of the boat in current is 3 miles per hour.
Let us represent the rate of boat in still water hence and rate of boat in current be y. Also, we know that speed = distance/time. Hence, keep the values in formula -
Converting mixed fraction to fraction, time = 3/2 hour
Time = 1.5 hour
1.5 (x + y) = 12 : equation 1
Divide the equation 1 by 3
0.5 (x + y) = 4 : equation 2
6 (x - y) = 12 : equation 3
Divide the equation 3 by 6
(x - y) = 2
x = 2 + y : equation 4
Keep the value of x from equation 4 in equation 2
0.5 (2 + y + y) = 4
1 + y = 4
y = 4 - 1
y = 3 miles/ hour
Keep the value y in equation 4 to get x
x = 2 + 3
x = 5 miles per hour
The rate in still water and current is 5 and 3 miles per hour.
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The complete question is-
It takes a boat 1 (1/2) hr to go 12 mi downstream, and 6 hr to return. Find the rate of the boat in still water and the rate of the current.
Penelope invested $89,000 in an account paying an interest rate of 6 1/4% compounded continuously. Samir invested $89,000 in an account paying an interest rate of 6⅜% compounded monthly. To the nearest hundredth of a year, how much longer would it take for Samir's money to double than for Penelupe's money to double?
Answers
Answer: -10.57
Step-by-step explanation:
Answer:
0.25 years
Step-by-step explanation:
Penelope invested $89,000 in an account paying an interest rate of 6⅜% compounded continuously.
To calculate the time it would take Penelope's money to double, use the continuous compounding interest formula.
[tex]\boxed{\begin{minipage}{8.5 cm}\underline{Continuous Compounding Interest Formula}\\\\$ A=Pe^{rt}$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\\phantom{ww}$\bullet$ $P =$ principal amount \\\phantom{ww}$\bullet$ $e =$ Euler's number (constant) \\\phantom{ww}$\bullet$ $r =$ annual interest rate (in decimal form) \\\phantom{ww}$\bullet$ $t =$ time (in years) \\\end{minipage}}[/tex]
As the principal amount is doubled, then A = 2P.
Given interest rate:
r = 6.375% = 0.06375Substitute A = 2P and r = 0.06375 into the continuous compounding interest formula and solve for t:
[tex]\implies 2P=Pe^{0.06375t}[/tex]
[tex]\implies 2=e^{0.06375t}[/tex]
[tex]\implies \ln 2=\ln e^{0.06375t}[/tex]
[tex]\implies \ln 2=0.06375t\ln e[/tex]
[tex]\implies \ln 2=0.06375t(1)[/tex]
[tex]\implies \ln 2=0.06375t[/tex]
[tex]\implies t=\dfrac{\ln 2}{0.06375}[/tex]
[tex]\implies t=10.872896949...[/tex]
Therefore, it will take 10.87 years for Penelope's investment to double.
[tex]\hrulefill[/tex]
Samir invested $89,000 in an account paying an interest rate of 6¹/₄% compounded monthly.
To calculate the time it would take Samir's money to double, use the compound interest formula.
[tex]\boxed{\begin{minipage}{8.5 cm}\underline{Compound Interest Formula}\\\\$ A=P\left(1+\frac{r}{n}\right)^{nt}$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\ \phantom{ww}$\bullet$ $P =$ principal amount \\ \phantom{ww}$\bullet$ $r =$ interest rate (in decimal form) \\ \phantom{ww}$\bullet$ $n =$ number of times interest is applied per year \\ \phantom{ww}$\bullet$ $t =$ time (in years) \\ \end{minipage}}[/tex]
As the principal amount is doubled, then A = 2P.
Given values:
A = 2PP = Pr = 6.25% = 0.0625n = 12 (monthly)Substitute the values into the formula and solve for t:
[tex]\implies 2P=P\left(1+\dfrac{0.0625}{12}\right)^{12t}[/tex]
[tex]\implies 2=\left(1+\dfrac{0.0625}{12}\right)^{12t}[/tex]
[tex]\implies 2=\left(1+0.005208333...\right)^{12t}[/tex]
[tex]\implies 2=\left(1.005208333...\right)^{12t}[/tex]
[tex]\implies \ln 2=\ln \left(1.005208333...\right)^{12t}[/tex]
[tex]\implies \ln 2=12t \ln \left(1.005208333...\right)[/tex]
[tex]\implies t=\dfrac{\ln 2}{12 \ln \left(1.005208333...\right)}[/tex]
[tex]\implies t=11.1192110...[/tex]
Therefore, it will take 11.12 years for Samir's investment to double.
[tex]\hrulefill[/tex]
To calculate how much longer it would take for Samir's money to double than for Penelope's money to double, subtract the value of t for Penelope from the value of t for Samir:
[tex]\begin{aligned}\implies t_{\sf Samir}-t_{\sf Penelope}&=11.1192110......-10.872896949...\\&= 0.246314066...\\&=0.25\; \sf years\;(nearest\;hundredth)\end{aligned}[/tex]
Therefore, it would take 0.25 years longer for Samir's money to double than for Penelope's money to double.
everyone pls answer the questions I posted they are urgent
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Answer:
unfortunately there's no questions to be answered
If the sides of a rectangle are in the ratio 3:4 and the length of the diagonal is 10 cm, find the length of the sides
Answers
Answer:
if the diagonal is 10 then the sides are 3*2 and 4*2 which is 6 and 8 respectively because the diagonal makes it a right angled triangle whereby the the 3,4,5 line steps in, so if the diagonal(hypotenuse) is 10 the 10/5 is 2 then you multiply both 3 and 4 by 2 and that gives you the length of two sides
Find the area under the standard normal distribution curve between z=0 and z=0. 98
Answers
The area under the standard normal distribution curve between z = 0 and z = 0.98 is:
0.8365 - 0.5000 = 0.3365
To find the area under the standard normal distribution curve between z = 0 and z = 0.98, we can use a standard normal distribution table or a calculator that can compute normal probabilities.
Using a standard normal distribution table, we can look up the area corresponding to a z-score of 0 and a z-score of 0.98 separately and then subtract the two areas to find the area between them.
The area under the standard normal distribution curve to the left of z = 0 is 0.5000 (by definition). The area under the curve to the left of z = 0.98 is 0.8365 (from the standard normal distribution table).
So the area under the standard normal distribution curve between z=0 and z=0.98 is approximately 0.3365.
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Two liters of the Gatorade cost $3.98. How much do 8 liters cost?
Answers
Answer:
$15.92
Step-by-step explanation:
We Know
2 liters of Gatorade cost $3.98
How much do 8 liters cost?
We take
3.98 x 4 = $15.92
So, 8 liters cost $15.92
Suppose a 4 is rolled on a number cube with sides numbered 1, 2, 3, 4, 5, and 6. The
complement of this event would be rolling a 1, 2, 3, 5, or 6. What is the probability of the
complement, written as a fraction in simplest form?
Answers
The probability of rolling any number other than 4 on a number cube with sides numbered 1, 2, 3, 4, 5, and 6 is 5/6, which can be written as a fraction in simplest form.
The complement of rolling a 4 on a number cube with sides numbered 1, 2, 3, 4, 5, and 6 is rolling any number other than 4, which includes rolling a 1, 2, 3, 5, or 6.
To find the probability of the complement, we need to add up the probabilities of rolling each of these numbers.
Since each number has an equal chance of being rolled, we can find the probability of rolling each number by dividing 1 by the total number of possible outcomes (which is 6, since there are six sides on the cube).
Then, we can add up the probabilities of rolling each of the five numbers in the complement:
P(rolling a 1, 2, 3, 5, or 6) = P(rolling a 1) + P(rolling a 2) + P(rolling a 3) + P(rolling a 5) + P(rolling a 6)
P(rolling any number other than 4) = 1 - P(rolling a 4)
P(rolling any number other than 4) = 1 - 1/6 = 5/6
Therefore, the probability of rolling any number other than 4 on a number cube with sides numbered 1, 2, 3, 4, 5, and 6 is 5/6, which can be written as a fraction in simplest form.
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Which expression represents the second partial sum for ? 2(0. 4) + 2(0. 4)2 2(0. 4)2 + 2(0. 4)3 2 + 2(0. 4) 0 + 2(0. 4)1
timed
Answers
The expression represents the second partial sum for 2(0. 4) + 2(0. 4)2 2(0. 4)2 + 2(0. 4)3 2 + 2(0. 4) 0 + 2(0. 4)1 is 0.8.
The second partial sum of a sequence refers to the sum of the first two terms of the sequence.
The given sequence is: 2(0.4) + 2(0.4)^2 + 2(0.4)^2 + 2(0.4)^3 + 2(0.4)^0 + 2(0.4)^1
To find the second partial sum, we simply add the first two terms of the sequence:
2(0.4) + 2(0.4)^2 = 0.8
Therefore, the expression that represents the second partial sum for the given sequence 2(0. 4) + 2(0. 4)2 2(0. 4)2 + 2(0. 4)3 2 + 2(0. 4) 0 + 2(0. 4)1 is 0.8.
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x^2+8x+16 What is the perfect factored square trinomial
Answers
Answer:
The perfect factored square trinomial that is equivalent to the expression x^2 + 8x + 16 is:
(x + 4)^2
To see why this is the case, you can expand the expression (x + 4)^2 using the FOIL method:
(x + 4)^2 = (x + 4) * (x + 4)
= x^2 + 4x + 4x + 16
= x^2 + 8x + 16
So, x^2 + 8x + 16 can be factored as (x + 4)^2, which is a perfect square trinomial.
what is the probability that a random point on AK will be on BE
Answers
The probability of the event BE falling on a random point AK is 4/11
What is the probability of an event?A probability event can be defined as a set of outcomes of an experiment. In other words, an event in probability is the subset of the respective sample space.
In this problem, we need to determine our sample space;
The sample space = 11
The number of favorable outcomes = 4
The probability of a random point on AK to be on BE will be;
P = 4 / 11
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An electronics retailer offers an optional protection plan for a mobile phone it sells. Customers can choose to buy the protection plan for \$100$100dollar sign, 100, and in case of an accident, the customer pays a \$50$50dollar sign, 50 deductible and the retailer will cover the rest of the cost of that repair. The typical cost to the retailer is \$200$200dollar sign, 200 per repair, and the plan covers a maximum of 333 repairs.
Let X be the number of repairs a randomly chosen customer uses under the protection plan, and let F be the retailer's profit from one of these protection plans. Based on data from all of its customers, here are the probability distributions of X and F:
X=\# \text{ of repairs}X=# of repairsX, equals, \#, start text, space, o, f, space, r, e, p, a, i, r, s, end text 000 111 222 333
F=\text{ retailer profit}F= retailer profitF, equals, start text, space, r, e, t, a, i, l, e, r, space, p, r, o, f, i, t, end text \$100$100dollar sign, 100 -\$50−$50minus, dollar sign, 50 -\$200−$200minus, dollar sign, 200 -\$350−$350minus, dollar sign, 350
Probability 0. 900. 900, point, 90 0. 70. 070, point, 07 0. 20. 020, point, 02 0. 10. 010, point, 01
Find the expected value of the retailer's profit per protection plan sold
Answers
Note that the expected value of the retailers profit is - $114. This means he made a loss.
How did we arrive at this ?To find the expected value we must proceed as follows
Expected Value - E(F) is
Probability of F - P(F)
= 100 x ($100 - $200) + (P(F) = $50) x ($50 - $200) + (P(F) = $ -200) x ( $ - 200 - $200) + (P(F) = $- 350) x ($ -350 $ 200)
= (0.9) x (-100) + (0.07 ) x (-150) + (0.01) x (-550) + (0.02) x (-400)
= - 90 - 10.5 - 5.5 -8
E(F) = $ -114
So it is right to state that the expected value of the retailer's profit per protection plan sold is -$114, which is a loss.
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Full Question:
An electronics retailer sells mobile phones with an optional protection plan for $100. In case of an accident, the customer pays a $50 deductible and the retailer covers the rest of the repair cost, which is typically $200 per repair. The protection plan covers a maximum of 333 repairs.
Let X be the number of repairs a randomly chosen customer uses under the protection plan, and let F be the retailer's profit from one of these protection plans. The probability distributions of X and F are:
X = number of repairs: 0 1 2 3
Probability: 0.90 0.07 0.02 0.01
F = retailer profit: $100-$50-$200-$350
Probability: 0.90 0.07 0.02 0.01
The task is to find the expected value of the retailer's profit per protection plan sold.
Help me pls this is my last try
2. What is the smallest positive degree angle measure equivalent to tan-¹ (0.724)?
42.2°
31.0°
44.6°
35.9°
Answers
Jose created a ball pit for his little sister to play in. he put 40 red balls, 55 purple balls, 45 yellow balls, and 60 green balls into the ball pit. while his sister is playing, one ball rolls out of the pit. what is the probability that the ball is red? 0.17 0.17 0.20 0.20 0.25 0.25 0.40
Answers
If he put 40 red balls, 55 purple balls, 45 yellow balls, and 60 green balls into the ball pit. while his sister is playing, one ball rolls out of the pit. Therefore, the probability that the ball that rolled out of the pit is red is 0.2.
The probability of selecting a red ball from the ball pit can be found by dividing the number of red balls by the total number of balls in the pit.
Total number of balls = 40 + 55 + 45 + 60 = 200
Probability of selecting a red ball = Number of red balls / Total number of balls
Probability of selecting a red ball = 40 / 200
Probability of selecting a red ball = 0.2
Therefore, the probability that the ball that rolled out of the pit is red is 0.2.
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(c) Katrina recorded the average rainfall amount, in inches, for two cities over the course of 6 months. City A: {5, 2. 5, 6, 2008. 5, 5, 3} City B: {7, 6, 5. 5, 6. 5, 5, 6} (a) What is the mean monthly rainfall amount for each city? (b) What is the mean absolute deviation (MAD) for each city? Round to the nearest tenth. (c) What is the median for each city?
Answers
a) The mean monthly rainfall amount for City A is 334.17 inches and for City B is 5.83 inches.
b) The MAD for City A is 464.28 inches and for City B is 0.46 inches.
c) The median for City A is 5 inches and for City B is 6 inches.
(a) To find the mean monthly rainfall amount for each city, we need to add up all the rainfall amounts and divide by the number of months:
For City A: (5 + 2.5 + 6 + 2008.5 + 5 + 3) / 6 = 334.17 inches
For City B: (7 + 6 + 5.5 + 6.5 + 5 + 6) / 6 = 5.83 inches
(b) To find the mean absolute deviation (MAD) for each city, we need to find the absolute deviations from the mean for each data point, then calculate the average of those absolute deviations:
For City A:
Mean = 334.17 inches
Absolute deviations from the mean: |5 - 334.17| = 329.17, |2.5 - 334.17| = 331.67, |6 - 334.17| = 328.17, |2008.5 - 334.17| = 1674.33, |5 - 334.17| = 328.17, |3 - 334.17| = 331.17
MAD = (329.17 + 331.67 + 328.17 + 1674.33 + 328.17 + 331.17) / 6 = 464.28 inches
For City B:
Mean = 5.83 inches
Absolute deviations from the mean: |7 - 5.83| = 1.17, |6 - 5.83| = 0.17, |5.5 - 5.83| = 0.33, |6.5 - 5.83| = 0.67, |5 - 5.83| = 0.83, |6 - 5.83| = 0.17
MAD = (1.17 + 0.17 + 0.33 + 0.67 + 0.83 + 0.17) / 6 = 0.46 inches
(c) To find the median for each city, we need to arrange the data points in order and find the middle value:
For City A: {2.5, 3, 5, 5, 6, 2008.5}
Median = 5 inches
For City B: {5, 5.5, 6, 6, 6.5, 7}
Median = 6 inches
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PYTHAGOREAN THEOREM!! HELP!! BRAINLIEST!! 20 POINTS!!
I know A and B! I need help with the rest!
Part A
The Pythagorean Theorem states that for any given right triangle, a^2+ b^2 = c^2.
Using the Pythagorean Theorem, what would be the relationship between the areas of the three squares (1, 2,and 3)?
Part B
Using squares 1, 2, and 3, and eight copies of the original triangle, you can create squares 4 and 5. What are the side lengths of square 4 and square 5 in terms of a and b? Do the two squares have the same area?
Part C
Write an expression for the area of square 4 by combining the areas of the four triangles and the two squares.
Part D
Write an expression for the area of square 5 by combining the area of the four triangles and one square.
Part E
Since the areas of square 4 and square 5 are the same, set the two expressions equal.
Part F
Which term is on both sides of the equal sign? Since it’s on both sides of the equal sign, you can cancel it out. What is the expression after canceling out the common term?
Part G
What does the equation show after you cancel out a common term?
Answers
The relationship between the areas of the three squares is that square A plus square B equals the area of square C.
What is Pythagorean Theory?The Pythagorean theorem is a fundamental idea in geometry that states that for any right-angled triangle, the square of the length of the longest side (opposite the right angle) is equal to the sum of the square of the lengths of the two remaining sides. This equation can be expressed as:
[tex]a^2 + b^2 = c^2[/tex]
Thus, the relationship between the areas of the three squares is that square A plus square B equals the area of square C.
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A dealer bought some radios for a total of $1,008. she gave away 6 radios as gifts, sold each of the rest for $14 more than she paid for each radio, and broke even. how many radios did she buy?
Answers
The dealer bought 42 radios.
How many radios did the dealer buy?Let x be the number of radios the dealer bought.
Let y be the price the dealer paid for each radio.
We know that the dealer bought x radios for a total of $1,008, so:
x * y = 1008
We also know that the dealer gave away 6 radios and sold the rest for $14 more than she paid for each radio, breaking even. This means that the total revenue from selling the remaining radios is equal to the total cost of buying them:
(x - 6) * (y + 14) = x * y
Simplifying this equation, we get:
xy + 14x - 6y - 84 = xy
14x - 6y = 84
7x - 3y = 42 (dividing by 2 on both sides)
Now we have two equations:
x * y = 1008
7x - 3y = 42
We can use substitution or elimination to solve for x and y. Let's use elimination by multiplying the second equation by y/3 and adding it to the first equation:
x * y + (7x - 3y) * (y/3) = 1008 + 42 * (y/3)
xy + 7xy/3 - y²/3 = 1008 + 14y
10xy/3 - y²/3 - 14y - 1008 = 0
Multiplying both sides by 3, we get:
10xy - y² - 42y - 3024 = 0
Now we can use the quadratic formula to solve for y:
y = (-b ± sqrt(b² - 4ac)) / 2a
where a = -1, b = -42, and c = -3024:
y = (-(-42) ± sqrt((-42)² - 4(-1)(-3024))) / 2(-1)
y = (42 ± sqrt(42² - 4*3024)) / 2
y = (42 ± 126) / 2
y = 84 or y = -42
Since the price of a radio cannot be negative, we can discard the second solution and conclude that y = 84.
Now we can solve for x using the first equation:
x * y = 1008
x * 84 = 1008
x = 12
Therefore, the dealer bought 12 radios.
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The cost C (in dollars) for the care and maintenance of a horse and carriage is C=15x+2000, where x is the number of rides. Write an equation for the revenue R in terms of the number of rides.
Answers
The equation for revenue R in terms of the number of rides x is given by R = px, where p is the amount charged per ride (in dollars).
The equation for the revenue R in terms of the number of rides can be derived by multiplying the number of rides with the amount charged per ride.
Let the amount charged per ride be p (in dollars).
Then, the equation for revenue R can be written as R = px.
Note that the amount charged per ride is not given in the problem. It can be assumed that the amount charged is a fixed amount for all the rides.
However, the equation for revenue can still be written in terms of the variable p as R = px.
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(1 point) Consider the function f(x, y) = xy + 33 – 48y. f har ? at (-43,0) f has at (0,4). a maximum a minimum a saddle some other critical point no critical point f ha: at (463,0). f has at (0,0). f has ? at (0, –4).
Answers
At point (-43,0), f has a maximum. At point (0,4), f has a minimum. At point (463,0), f has a maximum. At point (0,0), f may have a critical point. none of the given points are critical points of the function f (x, y) = xy + 33 - 48y.
Hi! To analyze the critical points of the function f(x, y) = xy + 33 - 48y, we first need to find the partial derivatives with respect to x and y:
fx = ∂f/∂x = y
fy = ∂f/∂y = x - 48
Now, we can analyze the given points:
1. (-43, 0)
At this point, fx = 0 and fy = -48. Since both partial derivatives are not equal to 0, this point is not a critical point.
2. (0, 4)
At this point, fx = 4 and fy = 0. Again, both partial derivatives are not equal to 0, so this is not a critical point.
3. (463, 0)
At this point, fx = 0 and fy = 463 - 48 = 415. Since both partial derivatives are not equal to 0, this is not a critical point.
4. (0, 0)
At this point, fx = 0 and fy = -48. Since both partial derivatives are not equal to 0, this is not a critical point.
5. (0, -4)
At this point, fx = -4 and fy = -48. Again, both partial derivatives are not equal to 0, so this is not a critical point.
In summary, none of the given points are critical points of the function f(x, y) = xy + 33 - 48y.
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Ava has two frogs. This is __
1
3 the number of frogs that Heather
has. How many frogs does Heather have? Draw a diagram to
represent the division. Then write and solve an equation.
Answers
The value of n which is the number of frogs Heather has is 6.
What is the number of frogs Heather has?The number of frogs Heather has is calculated as follows;
let the number of frogs Heather has = n
So Ava has 2 fogs, which is equal to 1/3 n.
The value of n which is the number of frogs Heather has is calculated as follows;
(1/3) n = 2
multiply both sides by 3;
n = 3 x 2
n = 6
The division using a diagram, is determined as;
0 0
I I
I I
I I
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Weights of erasers produced by a certain factory are known to follow the uniform distribution between 31. 5 g and 32. 3 g.
(a) (10 points) erasers produced by this factory are sold in packs of 45. A retailer randomly bought 200 packs. Find the probability that, for at least 15 packs, the average weight of the erasers in the pack is at least 31. 95 g.
(b) (10 points) each day, a quality control unit examines the erasers produced by this factory. The unit randomly chooses an eraser from the outputs of this factory and weighs it. This process is repeated 50 times. The unit then records the total number of erasers that were found to weigh at least 31. 7 g. (erasers with weights at least 31. 7 g are called "good" erasers)suppose this unit works for 42 consecutive days. Find the probability that, on average, it finds at least 37. 2 "good" erasers per day
Answers
a) The probability that, for at least 15 packs, the average weight of the erasers in the pack is at least 31.95 g is approximately 0.0384.
b) The probability that, on average, the unit finds at least 37.2 "good" erasers per day is approximately 0.3133.
a) To solve this problem, we need to use the central limit theorem. According to this theorem, the distribution of sample means becomes approximately normal, regardless of the shape of the population distribution, when the sample size is sufficiently large (usually, n >= 30). In this case, since the sample size is 45, we can assume that the distribution of sample means will be approximately normal.
Now, we need to find the probability that the average weight of at least 15 packs is at least 31.95 g. We can use the normal distribution to calculate this probability. We first calculate the z-score for this value as follows:
z = (31.95 - 31.9) / (0.163 / √(45)) = 1.77
Using a standard normal table or calculator, we can find the probability that a z-score is greater than or equal to 1.77. This probability is approximately 0.0384.
b) To solve this problem, we need to use the normal approximation to the binomial distribution. Since each eraser is either "good" or "bad", the number of "good" erasers that the unit finds each day follows a binomial distribution with parameters n = 50 and p = probability of finding a "good" eraser = (32.3 - 31.7)/(32.3 - 31.5) = 0.5.
Now, we need to find the probability that, on average, the unit finds at least 37.2 "good" erasers per day. We can use the normal distribution to calculate this probability. We first calculate the z-score for this value as follows:
z = (37.2 - 25) / 25 = 0.488
Using a standard normal table or calculator, we can find the probability that a z-score is greater than or equal to 0.488. This probability is approximately 0.3133.
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