To find the rate of change of the temperature difference between the two spacecraft, we need to first find the temperature at each spacecraft's position at time t=4.
For the first spacecraft, rt sin(t) = r4sin(4) and t=4, so its position is (4sin(4), 4, 0). Using the temperature function, we have T(4sin(4), 4, 0) = (4sin(4))(4)(5-2) = 48.08.
For the second spacecraft, rz cos(t) = r3cos(4) and t=-4/3, so its position is (3cos(4), -4/3, 7). Using the temperature function, we have T(3cos(4), -4/3, 7) = (3cos(4))(-4/3)(5-2) = -9.09.
Therefore, the temperature difference D between the two spacecraft at time t=4 is D = 48.08 - (-9.09) = 57.17.
To find the rate of change of D with respect to time, we use the Chain Rule. Let x = 4sin(t) and y = 4, so D = T(x, y, 0) - T(3cos(t), -4/3, 7). Then,
dD/dt = dD/dx * dx/dt + dD/dy * dy/dt
We already know that D = 48.08 - 9.09 = 57.17, so dD/dx = dT/dx = y(5-2x) = 4(5-2(4sin(4))) = -31.64.
We also have dx/dt = 4cos(4) and dy/dt = 0, since y is constant.
To find dD/dy, we take the partial derivative of T with respect to y, holding x and z constant: dT/dy = x(5-2y) = (4sin(4))(5-2(4)) = -28.16.
Putting it all together, we get:
dD/dt = dD/dx * dx/dt + dD/dy * dy/dt
= (-31.64)(4cos(4)) + (-28.16)(0)
= -126.56
Therefore, the rate of change of the temperature difference between the two spacecraft at time t=4 is -126.56.
Given the paths of the two spacecraft: r1(t) = (t sin(t), t, 0) and r2(t) = (t cos(t), -t, 7), and the temperature function T(x, y, z) = x * y * z^2, we want to determine the rate of change of the temperature difference D at time t=4 using the Chain Rule.
First, let's find the temperature for each spacecraft at time t:
T1(t) = T(r1(t)) = (t sin(t)) * t * 0^2
T1(t) = 0
T2(t) = T(r2(t)) = (t cos(t)) * (-t) * 7^2
T2(t) = -49t^2 cos(t)
Now, find the temperature difference D(t) = T2(t) - T1(t) = -49t^2 cos(t)
Next, find the derivative of D(t) with respect to t:
dD/dt = -98t cos(t) + 49t^2 sin(t)
Now, we need to evaluate dD/dt at t=4:
dD/dt(4) = -98(4) cos(4) + 49(4)^2 sin(4) ≈ -104.32
Thus, the rate of change of the temperature difference D at time t=4 is approximately -104.32 (in decimal notation, rounded to two decimal places).
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PJ conducts an experimental study on the effects of soft music during high stakes science testing. He randomly assigns students at the school. In one condition he does not provide music for testing while in the other group he does provide the music. He administers a pretest at the beginning of the year and a posttest at the end of the year. PJ's design is best described as a:
PJ's design is best described as a randomized controlled trial (RCT), which is a type of experimental study that randomly assigns participants to a control group or an intervention group.
In this case, the control group did not receive the soft music during high stakes science testing, while the intervention group did. By administering both a pretest and a posttest, PJ was able to measure any differences in performance between the two groups. The use of randomization helps to ensure that any differences observed between the groups can be attributed to the intervention (soft music) rather than to other factors that may have influenced the results. Overall, PJ's RCT design is a rigorous way to test the effects of a specific intervention on a particular outcome.
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A square based pyramid has a side length of 10 inches and a volume of 3300 inches^3. What is the height of the pyramid?
the height of the pyramid is 99 inches.
(explain)
To solve this problem, we can use the formula for the volume of a square pyramid which is:
Volume = (1/3) x (base area) x (height)
Since the base of our pyramid is a square with a side length of 10 inches, the base area would be:
Base area = (side length)^2 = 10^2 = 100 square inches
Substituting the values given in the problem, we get:
3300 = (1/3) x 100 x height
Multiplying both sides by 3, we get:
9900 = 100 x height
Dividing both sides by 100, we get:
height = 99 inches
Therefore, the height of the pyramid is 99 inches.
Consider this equation
(4x)^1/3 - x =0
The first step in solving this equation is to ____
The second step is to____
Solving this equation for x initially yields____
Checking the solutions shows that____
The first step in solving the equation is to isolate the radical term, and the second step is to eliminate the radical by raising both sides to the power of 3. Solving this equation for x initially yields x = 0, x = 2, and x = -2. Checking the solutions shows that it is not a real solution.
The first step in solving the equation (4x)[tex]^{1/3}[/tex] - x = 0 is to isolate the radical term by adding x to both sides of the equation. This gives us
(4x)[tex]^{1/3}[/tex]= x.
The second step is to raise both sides of the equation to the power of 3, which eliminates the radical. This results in 4x = x³.
Solving this equation for x initially yields x = 0, x = 2, and x = -2.
Checking the solutions shows that only x = 2 is a valid solution, as plugging it back into the original equation yields (4*2)[tex]^{1/3}[/tex] - 2 = 0. However, plugging in x = 0 results in a division by zero, and plugging in x = -2 yields a negative number under the radical, which is not a real solution.
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Question 20 of 20
Two numbers have a sum of 2 and a difference of 8. Write a system and solve it to identify the two numbers.
5 and 3
O 11 and -9
-5 and 7
O5 and-3
-5 and 7
Step-by-step explanation
lets try putting -5 and 7
-5 +7=2
and the difference is 8
i am not smart sorry
Pls respond quick | what is the value of the expression? c711
a)330
b)1,663,200
c)5040
d)7920
The value of the expression 11C7 is 330. This can be calculated using the formula for combinations, which is nCr = n!/r!(n-r)!, where n is the total number of objects and r is the number of objects being selected. So, the correct answer is A).
To calculate the value of the expression 11C7, we need to use the formula for combinations or binomial coefficients, which is
nCr = n! / (r! * (n-r)!)
where n is the total number of items, r is the number of items to be chosen, and ! denotes the factorial operation (the product of all positive integers up to n).
In this case, we have
n = 11 and r = 7
So, we can substitute these values into the formula
11C7 = 11! / (7! * (11-7)!)
= 11! / (7! * 4!)
= (11 * 10 * 9 * 8 * 7 * 6 * 5) / (4 * 3 * 2 * 1)
= 330
Therefore, the value of the expression 11C7 is 330. So, the correct answer is A).
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--The given question is incomplete, the complete question is given
"Pls respond quick | what is the value of the expression? ₁₁C₇
a)330
b)1,663,200
c)5040
d)7920"--
Carlos spots an airplane on radar that is currently approaching in a straight line, and that will fly directly overhead. the plane maintains a constant altitude of 7275 feet. carlos initially measures an angle of elevation of 20°
∘
to the plane at point aa. at some later time, he measures an angle of elevation of 37°
∘
to the plane at point bb. find the distance the plane traveled from point aa to point bb. round your answer to the nearest foot if necessary.
The distance the plane traveled from point A to point B is approximately y - x:
Distance = y - x
≈ 14046.99 feet - 20246.71 feet
≈ -6200.72 feet.
To find the distance the plane traveled from point A to point B, we can use trigonometry and the concept of similar triangles.
Let's denote the distance from point A to the plane as x, and the distance from point B to the plane as y. We are given the altitude of the plane (constant) as 7275 feet.
At point A, Carlos measures an angle of elevation of 20 degrees to the plane, and at point B, he measures an angle of elevation of 37 degrees to the plane.
Using trigonometry, we can set up the following equations:
tan(20 degrees) = 7275 / x,
tan(37 degrees) = 7275 / y.
We can rearrange these equations to solve for x and y:
x = 7275 / tan(20 degrees),
y = 7275 / tan(37 degrees).
Using a calculator, we can evaluate these expressions:
x ≈ 20246.71 feet,
y ≈ 14046.99 feet.
Therefore, the distance the plane traveled from point A to point B is approximately y - x:
Distance = y - x
≈ 14046.99 feet - 20246.71 feet
≈ -6200.72 feet.
Since the distance cannot be negative, we can round the absolute value of the result to the nearest foot:
Distance ≈ 6201 feet.
To find the distance the plane traveled from point A to point B, we can use trigonometry and the concept of similar triangles.
Let's denote the distance from point A to the plane as x, and the distance from point B to the plane as y. We are given the altitude of the plane (constant) as 7275 feet.
At point A, Carlos measures an angle of elevation of 20 degrees to the plane, and at point B, he measures an angle of elevation of 37 degrees to the plane.
Using trigonometry, we can set up the following equations:
tan(20 degrees) = 7275 / x,
tan(37 degrees) = 7275 / y.
We can rearrange these equations to solve for x and y:
x = 7275 / tan(20 degrees),
y = 7275 / tan(37 degrees).
Using a calculator, we can evaluate these expressions:
x ≈ 20246.71 feet,
y ≈ 14046.99 feet.
Therefore, the distance the plane traveled from point A to point B is approximately y - x:
Distance = y - x
≈ 14046.99 feet - 20246.71 feet
≈ -6200.72 feet.
Since the distance cannot be negative, we can round the absolute value of the result to the nearest foot:
Distance ≈ 6201 feet.
Therefore, the distance the plane traveled from point A to point B is approximately 6201 feet.
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A cylindrical shark tank with a height of 3 meters and a diameter of 15 meters holds 3 sharks. What is the
population density of the shark tank? (round answer to 4 decimal places)
The population density of the fish is 0.0061sharks/m²
What is population density?Population density is a measurement of population per unit land area. Therefore the population density can be expressed as;
population density = population/ area
The number of fish in the tank is 3
The area of the tank is given as!
A = 2πr( r+h)
h = 3meters
r = d/2 = 15/2 = 7.5
A = 2 × 3.14 × 7.5(7.5+3)
A = 47.1( 10.5)
A = 494.55 m²
Therefore population density = 3/494.55
= 0.0061sharks/m²
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find the exact values of the side lengths b & h
The value of the side lengths b and h in the right-angle triangle is [tex]3\sqrt{2} , and[/tex] 8.
What is a trigonometric ratio, exactly?Trigonometric ratios are the ratios of a right triangle's sides. The sine (sin), cosine (cos), and tangent are three often used trigonometric ratios. (tan).
The given figure is a right-angle triangle.
To find the value of b and h we need to apply the trigonometric ratio.
In the first triangle,
[tex]cos45 = \frac{adjacent}{hypotenuse}[/tex]
[tex]cos45 = \frac{b}{6}[/tex]
[tex]\frac{1}{\sqrt{2} } = \frac{b}{6}[/tex]
[tex]b = \frac{1}{\sqrt{2} } * 6[/tex]
[tex]b = 3\sqrt{2}[/tex]
In the second triangle
[tex]cos60 = \frac{adjacent}{hypotenuse} \\cos60 = \frac{4}{h} \\\frac{1}{2} = \frac{4}{h} \\h= 2 *4\\h = 8[/tex]
Therefore the value of the b and h is [tex]3\sqrt{2}[/tex] , and 8 respectively.
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A woman bought 130kg of tomatoes for 52. 0. She sold half of them at a profit of 30%. The rest of the tomatoes started to go bad. She then reduced the selling price per kg by 12%. Calculate
i. The new selling price per kg
ii. The percentage profit on the whole transaction if she threw away 5kg of bad tomatoes
(I) The new selling price per kilogram of the tomatoes is 0.4576.
(II) The percentage profit on the whole transaction is 24.77% if she threw away 5kg of bad tomatoes.
What is the new selling price?The new selling price is calculated as follows;
The cost per kilogram of the tomatoes is;
Cost per kg = Total cost / Total weight
Cost per kg = 52 / 130
Cost per kg = 0.4
Selling price per kg = Cost per kg + (Profit percentage x Cost per kg)
Selling price per kg = 0.4 + (0.3 x 0.4)
Selling price per kg = 0.52
The new selling price per kilogram is:
= Selling price per kg - (Reduction percentage x Selling price per kg)
= 0.52 - (0.12 x 0.52)
= 0.4576
The total revenue from selling the tomatoes is calculated as;
The woman sold half of the 130kg of tomatoes, = 130 / 2 = 65kg
Revenue = (amount sold x selling price per kg) + (amount left x new selling price per kg)
Revenue = (65 x 0.52) + (65 x 0.4576)
Revenue = 33.8 + 29.68
Revenue = 63.48
New total cost = Total cost / Total weight x (Total weight - Bad tomatoes)
New total cost = 0.4 x (130 - 5)
New total cost = 50.6
The profit on the whole transaction is calculated as;
Profit = Total revenue - New total cost
Profit = 63.48 - 50.6
Profit = 12.88
The profit percentage on the whole transaction is calculated as;
Profit percentage = (Profit / Total cost) x 100%
Profit percentage = (12.88 / 52) x 100%
Profit percentage = 24.77%
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Mrs. Dominguez has $9,400 to deposit into two different investment accounts. Mrs. Dominguez will deposit $3,500 into Account I, which earns 6. 5% annual simple interest She will deposit $5,900 into Account II, which earns 6% interest compounded annually. Mrs. Dominguez will not make any additional deposits or withdrawals. What is the total balance of these two accounts at the end of ten years? DE 10
Answer:
Step-by-step explanation:
The total balance of the two investment accounts at the end of ten years will be $16,564.08. To calculate the total balance of the two accounts at the end of ten years,
we need to use the formulas for simple interest and compound interest.
For Account I, the simple interest formula is:
I = Prt
where I is the interest earned, P is the principal (the amount deposited), r is the annual interest rate as a decimal, and t is the time in years.
Plugging in the values for Account I, we get:
I = (3500)(0.065)(10) = $2,275
So, after ten years, the balance in Account I will be:
B1 = P + I = 3500 + 2275 = $5,775
For Account II, the compound interest formula is:
A = P(1 + r/n)^(nt)
where A is the balance at the end of the time period, P is the principal, r is the annual interest rate as a decimal, n is the number of times the interest is compounded per year, and t is the time in years.
Plugging in the values for Account II, we get:
A = 5900(1 + 0.06/1)^(1*10) = $10,789.08
So, after ten years, the balance in Account II will be $10,789.08.
Therefore, the total balance of the two accounts at the end of ten years will be:
Total balance = Balance in Account I + Balance in Account II
= $5,775 + $10,789.08
= $16,564.08
In summary, by using the formulas for simple interest and compound interest, we can calculate that the total balance of the two investment accounts at the end of ten years will be $16,564.08.
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The total municipal debt for the state of Illinois can be represented as the exponential function M (t) = 41. 24(1. 052)^t
where M represents the total municipal debt for the state in billions of dollars and t is the number of years since 2000
Determine the statement that interprets the function M (t).
A) The total municipal debt in Illinois was $41. 24 million in 2000 and increases about 1. 052% each year.
B) The total municipal debt in Illinois was $43. 38 billion in 2000 and increases about 5. 2% each year.
C) The total municipal debt in Illinois was $39. 20 billion in 2000 and increases about 105. 2% each year.
D)The total municipal debt in inois was $41. 24 billion in 2000 and increases about 5. 2% each year
The correct statement that interprets the function M(t) is:
B) The total municipal debt in llinois was $43.38 billion in 2000 and increases about 5.2% each year.
According to the question the function M(t) =41.24(1.052)^t represents the total municipal debt for the state of llinois in billions of dollars, where t is the number of years since 2000.
The incorrect options are:
Option A is incorrect as the function does not represent an increase of 1.052% each year, but rather an increase of 5.2% each year (since 1.052=1+0.052)
Option C is also incorrect because it suggests an increase of 105.2% each year, which is not possible.
Option D is also incorrect because it states that the total municipal debt was $41.24 billion in 2000, and increases about 5.2% each year.
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A data set is normally distributed with a mean of 27 and a standard deviation of 3. 5. Find the z-score for a value of 25, to the nearest hundredth. Z-score =
If a data set is normally distributed with a mean of 27 and a standard deviation of 3. 5, the z-score for a value of 25 is -0.57.
To find the z-score for a value of 25 in a normally distributed data set with a mean of 27 and a standard deviation of 3.5, we use the formula:
z = (x - μ) / σ
where:
x = the given value (25)
μ = the mean (27)
σ = the standard deviation (3.5)
Plugging in the values, we get:
z = (25 - 27) / 3.5
z = -0.57
Rounding to the nearest hundredth, the z-score for a value of 25 is -0.57.
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Complete the statements by selecting the correct answer from each-down menu. Carl's transactions for a year are given in the table. His broker charged him $5 per trade. How much does Carl pay his broker?
Carl pays his broker a total of $200 in fees for the 40 trades he made during the year.
To calculate how much Carl pays his broker, we need to first determine how many trades he made in a year. Looking at the table, we can see that Carl made a total of 40 trades - 20 buys and 20 sells. Since each trade incurs a $5 charge, we can multiply the number of trades by the cost per trade to get the total amount paid to the broker.
40 trades x $5 per trade = $200 paid to the broker in a year
It's important to note that transaction costs can have a significant impact on investment returns, especially for small accounts, so it's important to consider these costs when making investment decisions. Some brokers may offer lower transaction fees or other incentives to attract clients, so it's worth shopping around and comparing fees before choosing a broker. Additionally, it may be more cost-effective to use a robo-advisor or invest in index funds or exchange-traded funds (ETFs) that have lower fees compared to actively managed funds.
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A clown made purple and green balloon animals at a party. He kept track of the requests.
What is the probability that a randomly selected balloon animal is green and is shaped like a dog?
The probability that a randomly selected balloon animal is green and shaped like a dog is 0.231.
What is the probability?The probability is found using the data table given below:
Purple and giraffe = 3; Purple and dog = 3; Green and giraffe = 4; Green and dog = 3
Out of the total number of balloon animals made, the number of green dog balloon animals is 3.
The probability of randomly selecting a green dog balloon animal is found using the formula:
Probability = (number of green dog balloon animals) / (total number of balloon animals)Probability = 3 / (3 + 3 + 4 + 3) = 3/13
Probability = 0.231
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3/4+(1/3 divided by 1/6) - (-1/2)
3/4 + (1/3 divided by 1/6) - (-1/2) when simplified give 3 1/4
How to determine this
3/4 + (1/3 divided by 1/6) - (-1/2)
3/4 + (1/3 ÷ 1/6) - (-1/2)
Using the rule of BODMAS
Whee B = Bracket
O = Order
D = Division
M = Multiplication
A = Addition
S = Subtraction
By removing the bracket
3/4 + 1/3 ÷ 1/6 + 1/2
By dividing
3/4 + 1/3 * 6/1 + 1/2
3/4 + 6/3 + 1/2
3/4 +2 + 1/2
By finding the LCM
The LCM is lowest common factor of the denominator which is 4
= [tex]\frac{3+8+2}{4}[/tex]
= 13/4
= 3 1/4
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Find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point yields a local min or max.
y = x^3 / x^2 + 1
The critical point x = 0 does not yield a local minimum or maximum. The function is always decreasing.
To find the critical points and intervals for the function y = x^3 / (x^2 + 1), we'll first find the derivative using the Quotient Rule:
y'(x) = [(x^2 + 1)(3x^2) - x^3(2x)] / (x^2 + 1)^2
y'(x) = (3x^4 + 3x^2 - 2x^4) / (x^2 + 1)^2
y'(x) = (x^2 - 2x^2) / (x^2 + 1)^2
Now, we'll find the critical points by setting the derivative equal to zero:
0 = (x^2 - 2x^2) / (x^2 + 1)^2
0 = x^2(1 - 2) / (x^2 + 1)^2
0 = -x^2 / (x^2 + 1)^2
This equation is equal to zero only when x = 0. So, the critical point is x = 0.
Next, we'll use the First Derivative Test to determine if the critical point yields a local min or max. To do this, we'll evaluate the sign of y'(x) to the left and right of x = 0.
1. Left of x = 0 (for example, x = -1):
y'(-1) = (-1)^2(1 - 2) / (-1^2 + 1)^2 = -1 / 1^2 = -1 (negative)
2. Right of x = 0 (for example, x = 1):
y'(1) = (1)^2(1 - 2) / (1^2 + 1)^2 = -1 / 2^2 = -1/4 (negative)
Since the derivative is negative on both sides of the critical point, the function is decreasing for all x. Thus, the critical point x = 0 does not yield a local minimum or maximum. The function is always decreasing.
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help I want to get this done
Answer:
j: 0, m: (-4)
Step-by-step explanation:
RECALL:
Rational function is the func. expressed by polynomials p(x) and q(x) as:
p(x)/q(x) where q(x) is non-zero
j(m+4) must be non zero, or
j(m+4)≠0
j≠0 and m+4≠0
j≠0 and m≠(-4)
EMERGENCY HELP NEEDED!!! WIL MARK BRAINLEST!!!
F (X) = X + 3
G (X) = 7X + 4
WHAT DOES (F + G) (X) EQUAL??
The solution to the composite function (f + g)(x) is: 8x + 7
How to solve Composite Functions?Composite functions are said to occur when the output of one function is used as the input of another. If we have a function f and another function g, it means that the function fg(x), said as “ f of g of x”, is the composition of the two functions.
Now, we are given two functions as:
f(x) = x + 3
g(x) = 7x + 4
Thus, we can say that:
(f + g)(x) = f(x) + g(x)
= x + 3 + 7x + 4
= 8x + 7
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Complete the eqaution of the line through (-8, -2) and (-4, 6)
Answer:
y = 2x + 14
Step-by-step explanation:
y = mx + b to write the equation, we need 2 things: the slope and the y-intercept
y = ___x + ____
Slope:
Change in y over the change in x. We find the change by subtracting. The y values are 6 and -2. The x values are -4 and -8
[tex]\frac{6- (-2)}{-4 -(-8)}[/tex] = [tex]\frac{6+2}{-4+8}[/tex] = [tex]\frac{8}4}[/tex] = 2
The slope is 2.
y-intercept:
Use either of the points given and the slope 2 to find the y-intercept. I am going to use the points(-4,6). I will use -4 for x and 6 for y given from the point
y = mx + b
6 = 2(-4) + b
6 = -8 + b Add 8 to both sides
14 = b
The y-intercept is 14.
y = 2x + 14
Helping in the name of Jesus.
Someone please help with this i need it really fast
Answer:
67. 50
Step-by-step explanation:
15x4.50= 67.5
Determine the boundedness and monotonicity of the following sequences. If possible, find the GLB and LUB, (n) {) 1-2 3n+1) 3
The sequence (n){(1-2)/(3^n+1) + 3} is bounded and decreasing. The GLB is 3 and the LUB is 1.
To determine the boundedness and monotonicity of the sequence, we can look at the limit as n approaches infinity.
Taking the limit of the sequence, we have:
lim(n→∞) [(1-2)/(3^n+1) + 3] = 3
This means that the sequence approaches a finite value as n gets larger, so the sequence is bounded.
Next, to check the monotonicity of the sequence, we can take the first derivative of the sequence with respect to n:
d/dn [(1-2)/(3^n+1) + 3] = [(2-1)(-ln3)(3^n+1)]/[(3^n+1)^2]
Simplifying, we get:
d/dn [(1-2)/(3^n+1) + 3] = (-ln3)/(3^n+1)^2
Since the derivative is negative for all n, the sequence is decreasing.
To find the GLB and LUB, we can use the fact that the sequence is decreasing and bounded. Since the sequence approaches 3 as n approaches infinity, 3 is the lower bound.
To find the upper bound, we can use the fact that the sequence is decreasing and start with the second term, which is 2. Therefore, the upper bound is 2. Since 1 < 2, we can conclude that the LUB is 1.
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The set of numbers 1 7 11 and 36 contains values for m what value of m makes the inequality 4m + 8 < 36 true
The value of m that makes the inequality 4m + 8 < 36 true is m = 1 for the set of numbers 1 7 11 and 36 contains values for m.
An inequality is a mathematical expression in which the values on the left side of an equation are not equal to the values on the right side, but instead are either greater than or less than the values on the right side.
To find the value of m that makes the inequality 4m + 8 < 36 true, given the set of numbers {1, 7, 11, 36},
Isolate the variable m in the inequality. Subtract 8 from both sides:Now, we know that the value of m should be less than 7. From the given set of numbers {1, 7, 11, 36}, only 1 is less than 7. Therefore, the value of m that makes the inequality true is m = 1.
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¿Como la gastronomia puede ayudarnos a convivir armoniosamente?
Gastronomy is a way of promoting understanding among different cultures, and of bringing people and traditions closer together.
The practise of choosing, preparing, presenting, and consuming exquisite cuisine. The foundation of gastronomy lies in the connections between food, culture, and tradition. Gastronomy has emerged through time as a more potent cultural force than language or other influences among the peoples of the world.
Molecular gastronomy is a relatively recent branch of science that studies the physical and chemical changes that take place during cooking. Molecular cuisine is the name of the new culinary movement based on this emerging discipline.
Nowadays, the world may be broken down into separate gastronomic zones, where different cuisines are popular and similar cooking techniques are used. Throughout much of Southeast Asia, rice is the main food. The abundant and creative use of spices to give meals an extra flavour is what makes Indian and Indonesian cuisine unique. The prevalent ingredient in Mediterranean recipes is olive oil.
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Complete question:
How can gastronomy help us live harmoniously?
Verify that the function f(x) = -4x^2 + 12x - 4ln x attains an absolute maximum and absolute minimum on
[1/2,2].
Find the absolute maximum and minimum values.
To verify that the function f(x) = -4x^2 + 12x - 4ln x attains an absolute maximum and absolute minimum on [1/2,2], we can use the Extreme Value Theorem.
First, we need to check if the function is continuous on the interval [1/2,2] and differentiable on the open interval (1/2,2).
The function is continuous on [1/2,2] because it is a polynomial and the natural logarithm function is continuous on its domain.
To check if it is differentiable on (1/2,2), we need to take the derivative:
f'(x) = -8x + 12 - 4/x
This is defined and continuous on the open interval (1/2,2).
Now we can find the critical points by setting f'(x) = 0:
-8x + 12 - 4/x = 0
Multiplying both sides by x and rearranging, we get:
-8x^2 + 12x - 4 = 0
Dividing by -4, we get:
2x^2 - 3x + 1 = 0
This factors as (2x - 1)(x - 1) = 0, so the critical points are x = 1/2 and x = 1.
We also need to check the endpoints of the interval:
f(1/2) = -4(1/4) + 6 - 4ln(1/2) = 2 - 4ln(1/2)
f(2) = -4(4) + 12(2) - 4ln(2) = 8 - 4ln(2)
Now we can compare the function values at the critical points and endpoints to find the absolute maximum and minimum:
f(1/2) = 2 - 4ln(1/2) ≈ 5.39
f(1) = -4(1) + 12(1) - 4ln(1) = 8
f(2) = 8 - 4ln(2) ≈ 0.31
So the absolute maximum value is 8, which occurs at x = 1, and the absolute minimum value is 0.31, which occurs at x = 2.
Therefore, the function f(x) = -4x^2 + 12x - 4ln x attains an absolute maximum and absolute minimum on [1/2,2], and the absolute maximum value is 8 and the absolute minimum value is 0.31.
To verify that the function f(x) = -4x^2 + 12x - 4ln(x) attains an absolute maximum and minimum on the interval [1/2, 2], we will first find its critical points by taking the first derivative and setting it to zero, and then evaluate the function at the critical points and endpoints.
The first derivative of f(x) is:
f'(x) = -8x + 12 - 4/x
Setting f'(x) to zero, we have:
-8x + 12 - 4/x = 0
Multiplying by x to remove the fraction, we get:
-8x^2 + 12x - 4 = 0
Dividing by -4, we have:
2x^2 - 3x + 1 = 0
Factoring, we get:
(x-1)(2x-1) = 0
This gives us the critical points x = 1 and x = 1/2.
Now, we evaluate f(x) at the critical points and endpoints:
f(1/2) = -4(1/2)^2 + 12(1/2) - 4ln(1/2)
f(1) = -4(1)^2 + 12(1) - 4ln(1)
f(2) = -4(2)^2 + 12(2) - 4ln(2)
Calculating these values, we get:
f(1/2) ≈ 5.386
f(1) = 4
f(2) ≈ -4
The absolute maximum value is ≈ 5.386 at x = 1/2, and the absolute minimum value is ≈ -4 at x = 2.
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Marlene wants to enclose her circular vegetable garden with fencing to keep rabbits from eating the vegetables. if the diameter of her garden is 14 feet, how much fencing will she need to buy if fencing is sold by the linear foot?
Marlene will need to buy approximately 44 feet of fencing to enclose her circular vegetable garden.
To determine the amount of fencing Marlene needs, we'll need to calculate the circumference of her circular vegetable garden. Here's a step-by-step explanation:
1. Given the diameter of the garden is 14 feet, we can find the radius by dividing the diameter by 2: Radius = Diameter / 2 = 14 feet / 2 = 7 feet.
2. Now, we will use the formula for the circumference of a circle, which is: Circumference = 2 * pi * radius.
3. Plug in the radius value we found in step 1: Circumference = 2 * pi * 7 feet.
4. Calculate the circumference: Circumference ≈ 2 * 3.1416 * 7 feet ≈ 43.98 feet.
5. Since fencing is sold by the linear foot, Marlene needs to buy approximately 44 linear feet of fencing to enclose her circular vegetable garden and keep the rabbits away.
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What is 133/14 simplify
Answer:
19/2
Step-by-step explanation:
133 = 7 × 19
14 = 7 × 2
133/14 = 19/2
Hence Simplified
Math 7 pap cdb 2 2020-2021 / 7 of 1
rectangle qrst is dilated with the origin as the center of dilation to create rectangle q'r's't'.
o'
r
q
which rule best represents the dilation applied to rectangle qrst to create rectangle q'r's't'?
o a. (x, y) - (3x, 3y)
o b. (x,y) - (1/3x, 1/3y)
o c. (x, y) - (x+3, y + 3)
o d. (x, y) - (x + 1/3, y + 1/3)
The correct answer is option (d) (x, y) → (x + 1/3, y + 1/3).
What is the rule that best represents the dilation applied to rectangle qrst to create rectangle q'r's't'?The dilation is a type of transformation that changes the size of the object, but not the shape. It can be represented by a rule in which each point of the original object is multiplied or divided by a constant factor. In this case, since the center of dilation is the origin, we can find the constant factor by dividing the coordinates of the corresponding points of the two rectangles.
Let's take the point Q(x, y) in rectangle QRST and its corresponding point Q'(x', y') in rectangle Q'R'S'T'. Since the dilation is centered at the origin, we have:
x' = k * x
y' = k * y
where k is the constant factor of dilation. We can find k by dividing the corresponding sides of the rectangles. For example, the length of QR is 4 units, and the length of Q'R' is 4/3 units. Therefore, k = 4/3.
Using this value of k, we can find the coordinates of any point in rectangle QRST that corresponds to a point in rectangle Q'R'S'T'. For example, point R(6, 2) in rectangle QRST corresponds to point R'(6 + 1/3, 2 + 1/3) = (20/3, 7/3) in rectangle Q'R'S'T'.
Hence, the correct answer is (x, y) → (x + 1/3, y + 1/3).
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One combine harvester can cut a 2450 square meters field in 5 hours. Another combines harvester can do the same job in 7 hours. What area can the two combines cut in 9 hours?
If combine harvester able to cut a 2450 square meters field in 5 hours and other one can do the same job in 7 hours then the area that the two combines cut in 9 hours is equals to the 7,560 square meters.
We have two harvester which can cut a area into different time.
The time taken by first harvester cuts into 2450 square meters field =5 hours.
The time taken by second harvester cuts into 2450 square meters field = 7 hours.
We have to determine the area can the two combines cut in 9 hours.
Here, total work = 2450 square meters
Work ability or efficiency of first harvester = 2450/5 = 490
Work efficiency of second harvester
= 2450/7= 350
Efficiency of both harvester in combine
= 350 + 490 = 840
So, area they cut into 9 hours = 840 × 9
=7560 square meters
Hence, required area is 7,560 square meters.
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Find each value or measure.
assume all lines that appear
to be tangent are tangent.
mztuv =
u
т.
539
v
s.
145º
The measure of angle TUV, given that line segment UT is tangent to circle V, and angle VTS is 145, is 55°.
Based on the information provided, you are looking to find the measure of angle TUV, given that line segment UT is tangent to circle V, and angle VTS is 145º.
Since UT is tangent to circle V, it means that angle UTV is a right angle (90º). Now, we know that the sum of the angles in a triangle is 180º. Therefore, to find the measure of angle TUV (m∠TUV), we can use the following formula:
m∠TUV + m∠UTV + m∠VTS = 180º
Substitute the given values:
m∠TUV + 90º + 145º = 180º
Solve for m∠TUV:
m∠TUV = 180º - 90º - 145º
m∠TUV = -55º
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625y2+400y-36+20z-z2
Answer:
The expression 625y^2 + 400y - 36 + 20z - z^2 can be rearranged and simplified as follows:
625y^2 + 400y - 36 + 20z - z^2
= (25y)^2 + 2(25y)(8) + 8^2 - 8^2 - 36 + 20z - z^2 (adding and subtracting (25y)(8) and 8^2 inside the parentheses)
= (25y + 8)^2 - (8^2 + 36) + 20z - z^2 (expanding the squared term and simplifying)
= (25y + 8)^2 - 100 + 20z - z^2 (simplifying)
Therefore, the simplified form of the expression is:
(25y + 8)^2 - 100 + 20z - z^2.
Note that this expression can also be written as:
(5y + 2)^2(5y - 12)^2 - (z - 10)(z + 10),
Using the difference of squares factorization. However, this is not necessarily simpler than the previous form, and it depends on the context and the purpose of the expression.
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