Solving for dz/dt, we get:
dz/dt ≈ 11.650 ft/sec.
So, after another minute, the distance between Claire and Anna is changing at a rate of approximately 11.650 ft/sec.
Hi there! To answer this question, we can use the Pythagorean theorem and implicit differentiation. Let x be the distance Claire runs east and y be the distance Anna runs north. After 1 minute, Claire has already run 12 * 60 = 720 ft. After another minute, x = 720 + 12t, and y = 7t.
Now, we can set up the Pythagorean theorem: x^2 + y^2 = z^2, where z is the distance between them. Substituting the expressions for x and y, we get (720 + 12t)^2 + (7t)^2 = z^2.
To find the rate at which the distance between them is changing (dz/dt), we need to differentiate both sides of the equation with respect to time, t:
2(720 + 12t)(12) + 2(7t)(7) = 2z(dz/dt).
Now, we can plug in the values for t = 2 minutes:
2(720 + 24)(12) + 2(14)(7) = 2z(dz/dt).
Simplifying, we get:
34560 + 392 = 2z(dz/dt).
After 2 minutes, Claire has run 12(120) = 1440 ft, and Anna has run 7(60) = 420 ft. Using the Pythagorean theorem, we can find z:
z = √(1440^2 + 420^2) ≈ 1500 ft.
Now we can find dz/dt:
34952 = 2(1500)(dz/dt).
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What is the tangent plane to z = ln(x−y) at point (3, 2, 0)?
The equation of the tangent plane to the surface z = ln(x - y) at the point (3, 2, 0) is x - y - z + 1 = 0.
To find the tangent plane to the surface z = ln(x - y) at the point (3, 2, 0), we can use the following steps
Find the partial derivatives of the surface with respect to x and y:
∂z/∂x = 1/(x - y)
∂z/∂y = -1/(x - y)
Evaluate these partial derivatives at the point (3, 2):
∂z/∂x (3, 2) = 1/(3 - 2) = 1
∂z/∂y (3, 2) = -1/(3 - 2) = -1
Use these values to find the equation of the tangent plane at the point (3, 2, 0):
z - f(3,2) = ∂z/∂x (3,2) (x - 3) + ∂z/∂y (3,2) (y - 2)
where f(x,y) = ln(x - y)
Plugging in the values we get:
z - 0 = 1(x - 3) - 1(y - 2)
Simplifying the equation, we get:
x - y - z + 1 = 0
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roger purchased a pair of pants for 34.50 and a new a new for 12.00 he had a 10% discount on his total purchased and paid 8.5% sales tax what was the total for rogers purchased
After the discount and the tax, the amount that Roger pays is $45.41
How to find the final price?We know that Roger purchased a pair of pants for 34.50 and a new a new for 12.00 he had a 10% discount on his total purchased and paid 8.5% sales tax, then the total cost before the discount and tax is:
C = 12.00 + 34.50 = 46.50
Now we apply the discount and the tax (as factors in a product) to get:
C' = 46.50*(1 - 0.1)*(1 + 0.085) = 45.41
That is the amouint that Roger pays for the two items.
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Determine the number of bricks, rounded to the nearest whole number, needed to complete the wall
The number of bricks, rounded to the nearest whole number, needed to complete the wall is 3,456 bricks.
To determine the number of bricks needed to complete a wall, you will need to know the dimensions of the wall and the size of the bricks being used. Let's say the wall is 10 feet high and 20 feet long, and the bricks being used are standard-sized bricks measuring 2.25 inches by 3.75 inches.
First, you'll need to convert the wall's dimensions from feet to inches. The wall is 120 inches high (10 feet x 12 inches per foot) and 240 inches long (20 feet x 12 inches per foot).
Next, you'll need to determine the number of bricks needed for each row. Assuming a standard brick orientation, you'll need to divide the length of the wall (240 inches) by the length of the brick (3.75 inches). This gives you 64 bricks per row (240/3.75).
To determine the number of rows needed, divide the height of the wall (120 inches) by the height of the brick (2.25 inches). This gives you 53.3 rows. Since you can't have a fraction of a row, round up to 54 rows.
To determine the total number of bricks needed, multiply the number of bricks per row (64) by the number of rows (54). This gives you 3,456 bricks. Rounded to the nearest whole number, the wall will need approximately 3,456 bricks to complete.
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14. If AB represents 50%, what is the length of a
line segment that is 100%?
Answer:
2*Ab Or AC
Step-by-step explanation:
No detail in question
In the figure, quadrilateral GERA is inscribed in circle P. TA is tangent to circle P at A, m∠REG = 78°, m AR ≅ 46°, and ER = GA. Find each measure
Someone please help will give brainliest
The measure of in quadrilateral GERA ∠GAR = 102° , ∠TAR = 23°, ∠GAN = 55° , m AG = 110° , m RE = 110° , m GE = 94°
∠REG = 78° , m AR = 46
The sum of the opposite angle of the quadrilateral is equal to 180°
∠REG + ∠GAR = 180°
∠GAR = 180 - ∠REG
∠GAR = 180 - 78
∠GAR = 102°
The tangent chord angle is half the intercept arc
∠TAR = 1/2 m AR
∠TAR = 1/2 ×46
∠TAR = 23°
The sum of straight angles is 180
m ∠GAN = 180 - (m ∠TAR + m ∠GAR )
m ∠GAN = 180 - (23 + 120)
m ∠GAN = 55°
The tangent chord angle is half the intercept arc
m AG = 2 m ∠GAN
m AG = 2(55)
m AG = 110°
as EG = GA
m RE = m GA
m RE = 110°
Complete angle sum = 360°
m GE = 360 - (m AG + m AR + m RE)
m GE = 360 - (110 + 46 + 110 )
m GE = 94°
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Hanson ate 68 out of g gumdrops. Write an expression that shows how many gumdrops Hanson has left
The expression g - 68 shows how many gumdrops Hanson has left after eating 68 out of g.
To find out how many gumdrops Hanson has left after eating 68 out of g, we need to subtract 68 from g. Therefore, the expression that shows how many gumdrops Hanson has left is:
g - 68
This expression represents the remaining gumdrops after Hanson has eaten 68 out of g. For example, if Hanson had 100 gumdrops before eating 68 of them, then the expression would be:
100 - 68 = 32
Therefore, Hanson would have 32 gumdrops left after eating 68 out of 100.
In summary, the expression g - 68 shows how many gumdrops Hanson has left after eating 68 out of g. The value of g represents the total number of gumdrops Hanson had before eating 68.
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The National Vital Statistics Reports for November 2011 states that U. S. Cesarean delivery rate for 2010 was about 32. 8%. Cesarean delivery is also called a "C-section. " It means the baby is not delivered in the normal way. The baby is surgically removed through an incision in the mother’s abdomen and uterus. Suppose this year a random sample of 100 births has 41 that are C-sections. Use the estimate from the NVS Report for 2011 as the population proportion, p, and the result from this year’s random sample to estimate the U. S. Cesarean delivery rate for this year with 95% confidence. (Be sure to check that a normal model is appropriate. )
The 95% confidence interval for the U.S. Cesarean delivery rate for this year is approximately (0.3314, 0.4886) or 33.14% to 48.86%.
How to find the delivery rate for a particular year using a sample and a population proportion estimate from a previous report?To estimate the U.S. Cesarean delivery rate for this year with 95% confidence using the provided information, we can construct a confidence interval for the population proportion.
Given:
Population proportion estimates from the NVS Report for 2011: p = 0.328 (32.8%)
Sample size: n = 100
Number of C-sections in the sample: x = 41
First, we need to check if a normal model is appropriate for the sample proportion. For this, we can verify if the sample size is sufficiently large and if both np and n(1-p) are greater than 10.
np = 100 * 0.328 = 32.8
n(1-p) = 100 * (1 - 0.328) ≈ 67.2
Since both np and n(1-p) are greater than 10, we can assume that the conditions for a normal model are met.
Now, we can calculate the confidence interval using the sample proportion and the critical value corresponding to a 95% confidence level.
Sample proportion (p-hat) = [tex]\frac{x }{ n}[/tex] =[tex]\frac{ 41 }{ 100 }[/tex]= 0.41
The critical value for a 95% confidence level can be obtained from a standard normal distribution or a Z-table. In this case, the critical value is approximately 1.96.
The margin of error (E) can be calculated as:
E = Z * [tex]\sqrt((\frac{p-hat * (1 - p-hat))} { n})[/tex]
E = 1.96 * [tex]\sqrt((\frac{0.41 * (1 - 0.41))}{ 100)}[/tex]
E ≈ 0.0786
Finally, we can construct the confidence interval by subtracting and adding the margin of error from the sample proportion.
Confidence interval = p-hat ± E
Confidence interval = 0.41 ± 0.0786
Therefore, the 95% confidence interval for the U.S. Cesarean delivery rate for this year is approximately (0.3314, 0.4886) or 33.14% to 48.86%.
Note: It's important to consider that this calculation assumes the sample is representative of the U.S. population and that the conditions for a normal model are satisfied. Additionally, the estimate from the NVS Report for 2011 is used as the population proportion.
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solve this trigonometric equation cos²x =3sin²x
Answer:
Step-by-step explanation:
cos²x =3sin²x subtract both sides by 3sin²x
cos²x - 3sin²x = 0 use identity cos²x+sin²x=1 => cos²x = 1-sin²x
substitute in
(1-sin²x)-3sin²x = 0 combine like terms
1-4sin²x=0 factor using difference of squares rule
(1-2sin x)(1+2sin x)=0 set each equal to 0
(1-2sin x)=0 (1+2sin x)=0
-2sinx = -1 2sinx= -1
sinx=1/2 sinx =-1/2
Think of the unit circle. When is sin x = ±1/2
at [tex]\pi /6, 5\pi /6, 7\pi /6, 11\pi /6[/tex]
This is from 0<x<2[tex]\pi[/tex]
a Open garbage attracts rodents. Suppose that the number of mice in a neighbourhood, I weeks after a strike by garbage collectors, can be approximated by the function P(t) = 2002. 10) a. How many mice are in the neighbourhood initially? b. How long does it take for the population of mice to quadruple? c. How many mice are in the neighbourhood after 5 weeks? d. How long does it take until there are 1000 mice? e. Find P' (5) and interpret the result.
a. There are 1000 mice in the neighborhood initially.
b. The population of mice never quadruple
c. After 5 weeks there are 18 mice in the neighborhood.
d. It takes 0 weeks for there to be 1000 mice.
e. The P' (5) is -96.86, indicates that after 5 weeks, the number of mice is declining at a pace of about 96.86 mice per week.
a. The initial number of mice in the neighborhood can be found by evaluating P(0):
P(0) = 2000/(1 + 10⁰/₁₀) = 2000/(1+1) = 1000
b. To find how long it takes for the population of mice to quadruple, we need to solve the equation:
P(t) = 4P(0)
2000/(1 + 10^(t/10)) = 4*1000
1 + 10^(t/10) = 1/4
10^(t/10) = -3/4
This equation has no real solutions, so the population of mice never quadruples.
c. To find how many mice are in the neighborhood after 5 weeks, we simply evaluate P(5):
P(5) = 2000/(1 + 10^(5/10)) = 2000/(1+100) = 18.18 (rounded to two decimal places)
Therefore, there are approximately 18 mice in the neighborhood after 5 weeks.
d. To find how long it takes until there are 1000 mice, we need to solve the equation:
P(t) = 1000
2000/(1 + 10^(t/10)) = 1000
1 + 10^(t/10) = 2
10^(t/10) = 1
t = 0
Therefore, there are 1000 mice in the neighborhood initially, so it takes 0 weeks for there to be 1000 mice.
e. To find P'(5), we first find the derivative of P(t):
P'(t) = -2000ln(10)/10 * 10^(t/10) / (1 + 10^(t/10))^2
Then we evaluate P'(5):
P'(5) = -2000ln(10)/10 * 10^(1/2) / (1 + 10^(1/2))^2 ≈ -96.86
This means that the population of mice is decreasing at a rate of approximately 96.86 mice per week after 5 weeks.
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Harold, Rhonda, and Brad added water to beakers in science class. The line plot shows the amount of water, in cups, that they added to each of 14 beakers.
In the given line plot, the data represents the amount of water, in cups, that Harold, Rhonda, and Brad added to each of 14 beakers in their science class.
A line plot is a way to represent data that involves marking a number line for each data point and placing an “X” above the number that represents the value of that data point.
The line plot shows that most of the beakers were filled with either 1 or 2 cups of water. Specifically, there are 5 beakers with 1 cup of water and 6 beakers with 2 cups of water. There are also 2 beakers with 3 cups of water and 1 beaker with 4 cups of water.
The line plot provides a visual representation of the data that allows the viewer to quickly understand the distribution of the data. By seeing that most of the data is clustered around 1 and 2 cups of water, one can infer that the students were likely instructed to add a specific amount of water to each beaker. However, the presence of a few outliers, such as the beaker with 4 cups of water, suggests that some of the students may have made errors in their measurements or not followed the instructions closely.
Overall, the line plot provides a quick and easy way to visualize the distribution of the data and identify any outliers or patterns in the data. It is a useful tool for representing small to medium-sized datasets and is commonly used in education, research, and data analysis.
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Answer:
if this is study island than the answer is:
All of the beakers with more than of a cup of water added to them were filled by Harold. Harold added a total of
4
cup(s) of water to his beakers.
All of the beakers with exactly of a cup of water added to them were filled by Rhonda. Rhonda added a total of
15/8 or 1 7/8
cup(s) of water to her beakers.
Brad filled the rest of the beakers. Brad added a total of
13/8 or 1 5/8
cup(s) of water to his beakers.
Step-by-step explanation:
Algebra 2 question need help.
Answer:
c
Step-by-step explanation:
What are your chances of winning a raffle in which 325 tickets have been sold, if you haveone ticket?
Your chances of winning a raffle with one ticket out of 325 sold is approximately 0.31% or 1 in 325.
The probability of winning a raffle is determined by dividing the number of tickets you have by the total number of tickets sold. In this case, since there are 325 tickets sold and you have only one ticket, your chances of winning are 1 in 325, which is equivalent to a probability of approximately 0.31%.
This means that you have a very low chance of winning, but it's not impossible. However, the more tickets you have, the greater your chances of winning will be. It's important to remember that winning a raffle is a matter of luck and chance, and not a guaranteed outcome.
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help pls!
Use unit multipliers to convert 123 pounds per mile to ounces per centimeter.
There are 5,280 feet in 1 mile. There are 16 ounces in 1 pound. There are approximately 2.54 cm in 1 inch.
Enter your answer as a decimal rounded to the nearest hundredth. Just enter the number.
The conversion is given as follows:
123 pounds per mile = 0.01 ounces per cm.
How to obtain the conversion?The conversion is obtained applying the proportions in the context of the problem.
There are 16 ounces in 1 pound, hence the number of ounces in 123 pounds is given as follows:
123 x 16 = 1968 ounces.
There are 5,280 feet in 1 mile, 12 inches in one feet and 2.54 cm in one inch, hence the number of cm is given as follows:
5280 x 12 x 2.54 = 160934.4 cm.
Hence the rate is given as follows:
1968/160934.4 = 0.01 ounces per cm.
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Please help factor this expression completely, then place the factors in the proper location on the grid.
1/8 x^3-1/27 y^3
will mark brainly
Using cubes formula the factored expression is given as:
1/8x^3 - 1/27y^3 = (1/2x - 1/3y)(1/4x^2 + 1/6xy + 1/9y^2)
To factor the expression [tex]1/8x^3 - 1/27y^3[/tex], we can utilize the difference of cubes formula, which states that the difference of two cubes can be factored as the product of their binomial factors.
In our given expression, we have[tex](1/8x^3 - 1/27y^3).[/tex] We can identify[tex]a^3 as (1/2x)^3 and b^3 as (1/3y)^3.[/tex]
Applying the difference of cubes formula, we get:
[tex](1/8x^3 - 1/27y^3) = (1/2x - 1/3y)((1/2x)^2 + (1/2x)(1/3y) + (1/3y)^2)[/tex]
Simplifying the expression within the second set of parentheses, we have:
[tex](1/8x^3 - 1/27y^3) = (1/2x - 1/3y)(1/4x^2 + 1/6xy + 1/9y^2)[/tex]
Therefore, the factored form of the expression 1/8x^3 - 1/27y^3 is given by (1/2x - 1/3y)(1/4x^2 + 1/6xy + 1/9y^2). This represents the product of the binomial factors resulting from the application of the difference of cubes formula.
To factor the expression 1/8x^3 - 1/27y^3, we can use the difference of cubes formula, which states that:
[tex]a^3 - b^3 = (a - b)(a^2 + ab + b^2)[/tex]
Applying this formula, we get:
1/8x^3 - 1/27y^3 = (1/2x - 1/3y)(1/4x^2 + 1/6xy + 1/9y^2)
Therefore, the expression is completely factored as:
[tex]1/8x^3 - 1/27y^3 = (1/2x - 1/3y)(1/4x^2 + 1/6xy + 1/9y^2)[/tex]
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If AM=25CM, MC=20CM, MN=30CM, NC=35CM. What is the scale factor
The scale factor is 7/5 or 1.4.
f AM=25CM, MC=20CM, MN=30CM, NC=35CM.find scale factor
In order to determine the scale factor, we need to compare the corresponding sides of two similar figures. Let's begin by drawing a diagram to represent the given information:
M ------- N
/ \
/ \
A ---------------- C
<-----25cm----->
<-----20cm-----> <-----35cm----->
From the diagram, we see that triangle AMC is similar to triangle CNC, since they share angle C and have proportional sides:
Scale factor = corresponding side length in triangle CNC / corresponding side length in triangle AMC
We can calculate the scale factor by comparing the lengths of the corresponding sides:
Scale factor = NC / AM
Scale factor = 35 cm / 25 cm
Scale factor = 7 / 5
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(−2x−1)(−3x 2 +6x+8)
Find the volume of the figure.
Answer:
22(15)(12) + (1/2)(22)(10)(15) = 5,610 cm^2
Calculate the interest and total value on a $6,300 deposit for 8 years at a compound interest rate of 4. 5%
The interest is $2,659.23 and the total value is $8,959.23.
What is compound interest?
The interest that is calculated using both the principal and the interest that has accrued during the previous period is called compound interest. It differs from simple interest in that the principal is not taken into account when determining the interest for the subsequent period with simple interest.
Here the given principal P = $6300
Number of years = 8
Rate of interest = 4.5% = 4.5/100 = 0.045
Now using compound interest formula then,
=> Amount = [tex]P(1+r)^{t}[/tex]
=> Amount = 6300[tex](1+0.045)^8[/tex]
=> Amount = [tex]6300(1.045)^8[/tex]
=> Amount = $8,959.23
Then Interest = Amount - Principal
=> Interest = $8,959.23 - $6300 = $2,659.23.
Hence the interest is $2,659.23 and the total value is $8,959.23.
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A medical researcher is studying the effects of a drug on blood pressure. Subjects in the study have their blood pressure taken at the beginning of the study. After being on the medication for 4 weeks, their blood pressure is taken again. The change in blood pressure is recorded and used in doing the hypothesis test.
Change: Final Blood Pressure - Initial Blood Pressure
The researcher wants to know if there is evidence that the drug affects blood pressure. At the end of 4 weeks, 36 subjects in the study had an average change in blood pressure of 2. 4 with a standard deviation of 4. 5.
Find the
p
-value for the hypothesis test
The p-value for the hypothesis test is 0.04. This means that if the null hypothesis is true
To find the p-value, we need to conduct a hypothesis test.
The null hypothesis is that there is no difference in blood pressure before and after taking the medication:
H0: μd = 0
The alternative hypothesis is that there is a difference in blood pressure before and after taking the medication:
Ha: μd ≠ 0
where μd is the population mean difference in blood pressure before and after taking the medication.
We are given that the sample size is n = 36, the sample mean difference is ¯d = 2.4, and the sample standard deviation is s = 4.5.
We can calculate the t-statistic as:
t = (¯d - 0) / (s / sqrt(n)) = (2.4 - 0) / (4.5 / sqrt(36)) = 2.13
Using a t-distribution table with 35 degrees of freedom (df = n - 1), we find that the two-tailed p-value for t = 2.13 is approximately 0.04.
Therefore, the p-value for the hypothesis test is 0.04. This means that if the null hypothesis is true (i.e., if there is really no difference in blood pressure before and after taking the medication), there is a 4% chance of observing a sample mean difference as extreme or more extreme than 2.4. Since this p-value is less than the significance level of 0.05, we reject the null hypothesis and conclude that there is evidence that the drug affects blood pressure.
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Garden plots in the Portland Community Garden are rectangles
limited to 45 square meters. Christopher and his friends want a plot
that has a width of 7.5 meters. What length will give a plot that has
the maximum area allowed?
The length should be less than or equal to 6 meters in order to have a plot with the maximum area allowed (45 square meters) when the width is 7.5 meters.
To find the length that will give a plot with the maximum area allowed, we can use the formula for the area of a rectangle:
Area = Length × Width
The width is given as 7.5 meters, and the area should not exceed 45 square meters.
Let's denote the length as L.
We want to maximize the area, so we need to find the value of L that satisfies the condition Area ≤ 45 and gives the largest possible area.
Substituting the given values into the area formula, we have:
Area = L × 7.5
Since the area should not exceed 45 square meters, we can write the inequality:
L × 7.5 ≤ 45
To find the maximum value of L, we can divide both sides of the inequality by 7.5:
L ≤ 45 / 7.5
Simplifying the right side:
L ≤ 6
Therefore, the length should be less than or equal to 6 meters in order to have a plot with the maximum area allowed (45 square meters) when the width is 7.5 meters.
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point (4, -13) lies on the graph of the equation y = kx + 7
what is value of k?
Answer:
-5
Step-by-step explanation:
(4, -13) = (x, y)
y = kx + 7
-13 = k(4) + 7
4k = -13-7
4k = -20
k = -5
#CMIIWPLEASE HELPPPPPPPP
PLEASE IMBEGGING
The area under the curve at the given points is 3.758 sq.units.
What is the area under the curve?The area under the curve at the given points is calculated as follows;
y = -3/x ; (-7, -2)
To find the area under the curve y = -3/x between x = -7 and x = -2, we need to integrate the function from x = -7 to x = -2.
∫[-7,-2] (-3/x) dx
= [-3 ln|x|]_(-7)^(-2)
= [-3 ln|-2| - (-3 ln|-7|)]
= [-3 ln(2) + 3 ln(7)]
= 3 ln(7/2)
= 3.758 sq.units
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Max's niece pushed a playground merry-go-round so that it travels 4. 5 feet along the
curve. The radius of the merry-go-round is 5 feet. Find, to the nearest degree, the
central angle.
The central angle is approximately 51.6 degrees.
How to find the Arc length of a central angle?To solve this problem, we can use the formula for arc length of a circle:
arc length = θ × r
where θ is the central angle in radians, and r is the radius of the circle.
We know that the arc length is 4.5 feet and the radius is 5 feet. So we can rearrange the formula to solve for θ:
θ = arc length / r
θ = 4.5 / 5
θ = 0.9 radians
To find the central angle in degrees, we can convert radians to degrees by multiplying by 180/π:
θ = 0.9 × (180/π)
θ ≈ 51.6 degrees
Therefore, the central angle is approximately 51.6 degrees.
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Furnace repair bills are normally distributed with a mean of 264 dollars and a standard deviation of 30 dollars. if 144 of these repair bills are randomly selected, find the probability that they have a mean cost between 264 dollars and 266 dollars.
Answer is the probability that 144 furnace repair bills have a mean cost between 264 dollars and 266 dollars is approximately 0.2881 or 28.81%
The distribution of the sample mean of furnace repair bills will also be normally distributed with a mean of 264 dollars and a standard deviation of 30/sqrt(144) = 2.5 dollars (by the Central Limit Theorem).
We need to find the probability that the sample mean falls between 264 and 266 dollars:
z1 = (264 - 264) / 2.5 = 0
z2 = (266 - 264) / 2.5 = 0.8
Using a standard normal distribution table or calculator, we can find the area under the curve between z1 and z2:
P(0 ≤ Z ≤ 0.8) = 0.2881
Therefore, the probability that 144 furnace repair bills have a mean cost between 264 dollars and 266 dollars is approximately 0.2881 or 28.81%.
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BRANLIEST!!
Three coins are tossed. Let the event H = all Heads and the event K = at least one Heads.
1. 7/8 P(K) =
2. 1/7 The probability that the outcome is all heads if at least one coin shows a head
3. 1/8 P(H∩K) =
The probability that the outcome is all heads if at least one coin shows a head is 8/49.
How to find the probability?To solve these problems, we'll use the basic principles of probability.
The probability of an event K (at least one head) can be calculated by subtracting the probability of the complement of K (no heads) from 1.
Since the coins can either show all heads or not, the complement of K is the event of no heads, which is denoted as T (tails for all coins). Therefore, we have:
P(K) = 1 - P(T)
Each coin toss is independent, and the probability of getting tails on a single toss is 1/2. Since there are three coins tossed independently, we multiply the probabilities together:
P(T) = ([tex]\frac{1}{2}[/tex]) * ([tex]\frac{1}{2}[/tex]) * ([tex]\frac{1}{2}[/tex]) = [tex]\frac{1}{8}[/tex]
Substituting this into the equation for P(K):
P(K) = 1 - P(T) = 1 - [tex]\frac{1}{8}[/tex] = [tex]\frac{7}{8}[/tex]
So, the probability of event K (at least one head) is [tex]\frac{7}{8}[/tex].
The probability that the outcome is all heads if at least one coin shows a head can be calculated using conditional probability. We want to find P(H | K), which represents the probability of event H (all heads) given event K (at least one head).
The formula for conditional probability is:
P(H | K) = [tex]\frac{P(H \∩ K) }{ P(K)}[/tex]
To find P(H∩K), we need to determine the probability of the intersection of events H and K (i.e., the probability of getting all heads and at least one head).
Since H is a subset of K (if all coins show heads, then at least one head is shown), we have:
P(H∩K) = P(H)
Therefore, P(H∩K) is the same as P(H). According to the problem, P(H) = [tex]\frac{1}{7}[/tex].
Now, substituting P(H∩K) = P(H) and P(K) = [tex]\frac{7}{8}[/tex] into the conditional probability formula:
P(H | K) = [tex]\frac{P(H\∩K) }{ P(K)}[/tex] = ([tex]\frac{1}{7}[/tex]) / ([tex]\frac{7}{8}[/tex]) = ([tex]\frac{1}{7}[/tex]) * ([tex]\frac{8}{7}[/tex]) = [tex]\frac{8}{49}[/tex]
So, the probability that the outcome is all heads if at least one coin shows a head is [tex]\frac{8}{49}[/tex].
To summarize:
P(K) = [tex]\frac{7}{8}[/tex]
P(H | K) = [tex]\frac{8}{49}[/tex]
P(H∩K) = [tex]\frac{1}{7}[/tex]
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If y, p and q vary jointly and p is 14 when y and q are equal to 2, determine q when p and y are equal to 7
In the given question, if y, p and q vary jointly and p is 14 when y and q are equal to 2 and p and y are equal to 7, we get q is equal to 14 using the joint variation formula.
To solve this problem, we need to use the formula for joint variation, which states that y, p, and q vary jointly if there exists a constant k such that ypk = kq.
In this case, we know that when y=2 and q=2, p=14. So we can set up the equation: 2*14*k = 2kq
Simplifying this, we get: 28k = 2kq
Dividing both sides by 2k, we get: 14 = q
So when p=7 and y=7, we can use the same equation: 7*14*k = 7kq
Simplifying this, we get: 98k = 7kq
Dividing both sides by 7k, we get: q = 14
Therefore, when p and y are equal to 7, q is equal to 14.
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Shapes A and B are similar.
a) Calculate the scale factor from shape A to shape B.
b) Find the value of w.
Give each answer as an integer or as a fraction in its simplest form.
4 cm
7 cm
A
12 cm
3 cm
w cm
B
9 cm
Determine the vector equation of each of the following planes.
b) the plane containing the two intersecting lines r= (4,7,3) + t(2,4,3) and r= (-1,-4,6) + s(-1,-1,3)
To find the vector equation of the plane containing the two intersecting lines, we can first find the normal vector of the plane by taking the cross product of the direction vectors of the two lines. The normal vector will be orthogonal to both direction vectors and thus will be parallel to the plane.
Direction vector of the first line: (2, 4, 3)
Direction vector of the second line: (-1, -1, 3)
Taking the cross product of these two vectors, we get:
(2, 4, 3) x (-1, -1, 3) = (9, -3, -6)
This vector is orthogonal to both direction vectors and thus is parallel to the plane. To find the vector equation of the plane, we can use the point-normal form of the equation, which is:
N · (r - P) = 0
where N is the normal vector, r is a point on the plane, and P is a known point on the plane. We can choose either of the two given points on the intersecting lines as the point P.
Let's use the point (4, 7, 3) on the first line as the point P. Then the vector equation of the plane is:
(9, -3, -6) · (r - (4, 7, 3)) = 0
Expanding and simplifying, we get:
9(x - 4) - 3(y - 7) - 6(z - 3) = 0
Simplifying further, we get:
9x - 3y - 6z = 0
Dividing by 3, we get:
3x - y - 2z = 0
Therefore, the vector equation of the plane containing the two intersecting lines is:
(3, -1, -2) · (r - (4, 7, 3)) = 0
or equivalently,
3x - y - 2z = 0.
Write a derivative formula for the function.
f(x) = (4 ln(x))ex
The derivative formula for the function is f'(x) = 4ex(1/x + ln(x)).
How to determined the function by differentiation?To find the derivative of the function f(x) = (4 ln(x))ex, we can use the product rule and the chain rule of differentiation.
Let g(x) = 4 ln(x) and h(x) = ex. Then, we have:
f(x) = g(x)h(x)
Using the product rule, we get:
f'(x) = g'(x)h(x) + g(x)h'(x)
Now, we need to find g'(x) and h'(x):
g'(x) = 4/x (since the derivative of ln(x) with respect to x is 1/x)
h'(x) = ex
Substituting these back into the formula for f'(x), we get:
f'(x) = (4/x)ex + 4 ln(x)ex
Simplifying this expression, we get:
f'(x) = 4ex(1/x + ln(x))
Therefore, the derivative formula for the function f(x) = (4 ln(x))ex is:
f'(x) = 4ex(1/x + ln(x)).
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12
select the correct number from each drop-down menu to complete the equation
7
2 +
+ b
a
2
-2
The completed equation is:
2 + 7 = a - 2
a = 11.
We are given the following equation:
2 + b = a - 2
We need to select the correct number from the drop-down menu to complete the equation.
From the first drop-down menu, we select 7.
2 + 7 = 9
From the second drop-down menu, we select 2.
2 + b = 9 - 2
2 + b = 7
Subtracting 2 from both sides, we get:
b = 5
Therefore, from the third drop-down menu, we select 5.
So, the completed equation is:
2 + 7 = 5 - 2
9 = 3
This is not a true statement, so there must be an error in one of our selections. Upon closer inspection, we can see that the correct number to select from the first drop-down menu is 5, not 7.
2 + 5 = 7
Now, substituting 5 for b in the original equation, we get:
2 + 5 = a - 2
7 + 2 = a
a = 9
Therefore, from the third drop-down menu, we select 9.
So, the completed equation is:
2 + 5 = 9 - 2
7 = 7
This is a true statement, so we have selected the correct numbers to complete the equation.
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