A block of ice in the shape of a cube melts uniformly maintaining its shape. The volume of a cube given a side length is given by the formula V = S^3. At the moment S = 2 inches, the volume of the cube is decreasing at a rate of 5 cubic inches per minute. What is the rate of change of the side length of the cube with respect to time, in inches per minute, at the moment when S = 2 inches?

A. -5/12
B. 5/12
C. -12/5
D. 12/5

To find the antiderivative, we integrate each term separately:

V = π ∫[0, 4] ([tex]y^2[/tex] - [tex]y^{3/2[/tex] + [tex]y^{4/16[/tex]) dy

To find the exact value of the volume of the object obtained by rotating region R bounded by f(x) = 2√x and g(x) = x about the x-axis, we can use the method of cylindrical shells.

First, let's find the points of intersection between the two functions:

2√x = x

Squaring both sides:

4x = [tex]x^2[/tex]

Rearranging and factoring:

[tex]x^2[/tex] - 4x = 0

x(x - 4) = 0

x = 0 or x = 4

So, the points of intersection are (0, 0) and (4, 4).

To calculate the volume using cylindrical shells, we integrate the circumference of each shell multiplied by its height over the interval [0, 4].

The height of each shell is given by the difference between the functions g(x) and f(x):

h(x) = g(x) - f(x) = x - 2√x

The circumference of each shell is given by 2πx.

Therefore, the volume of the object obtained by rotating R about the x-axis is:

V = ∫[0, 4] 2πx * (x - 2√x) dx

Simplifying the integral:

V = 2π ∫[0, 4] ([tex]x^2[/tex] - 2x√x) dx

V = 2π ∫[0, 4] ([tex]x^2[/tex] - [tex]2x^{(3/2)[/tex]) dx

To find the antiderivative, we integrate each term separately:

V = 2π [ (1/3)[tex]x^3[/tex] - (2/5)[tex]x^{(5/2)[/tex] ] evaluated from 0 to 4

V = 2π [ (1/3)([tex]4^3[/tex]) - (2/5)([tex]4^{(5/2)[/tex]) ] - 2π [ (1/3)([tex]0^3[/tex]) - (2/5)([tex]0^{(5/2)[/tex]) ]

V = 2π [ (64/3) - (32/5) ]

V = 2π [ (320/15) - (96/15) ]

V = 2π [ 224/15 ]

V = (448π/15)

Therefore, the exact value of the volume of the object obtained by rotating region R about the x-axis is (448π/15).

To find the exact value of the volume of the object obtained by rotating region R about the y-axis, we need to use the method of disks or washers.

Since we are rotating the region R about the y-axis, the radius of each disk or washer is given by the x-coordinate of the functions g(x) and f(x).

The x-coordinate of g(x) is x = y, and the x-coordinate of f(x) is

x = [tex](y/2)^2[/tex]

= [tex]y^{2/4[/tex]

So, the radius is given by the difference between y and [tex]y^{2/4[/tex].

Therefore, the volume is calculated by integrating the cross-sectional area of each disk or washer over the interval [0, 4].

The cross-sectional area is given by π(radius)^2.

V = ∫[0, 4] π[[tex](y - y^{2/4})^2[/tex]] dy

Simplifying the integral:

V = π ∫[0, 4] ([tex]y^2 - y^{3/2} + y^{4/16[/tex]) dy

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Answer:

I think they charge a flat rate of $8 for a vehicle and $3 per person on top of that.

Step-by-step explanation:

There could be a multitude of rules, but let's try to find a simple one.

I would start by assuming that the park charges a fee for a vehicle and then for each person. Let's name those:

a - flat fee for a vehicle

b - fee per person

then a car with x people in it would be charged

a + bx

Let's see if the fees fit this assumption:

for x = 2

a + b*2 = $14

for x = 4

a + b*4 = $20

for x = 8

a + b*8 = $32

If so, we can subtract the second equation from the first and get

a + b*4 - a - b*2 = $20 - $14

b*2 = $6

b = 3$

a + b*2 = $14

a + $3*2 = a + $6 = $14

a = $8

so the first 2 equations gave us a = $8 and b = $3.

Let's see if these values fit the 3rd equation

a + b*8 = $32

$8 + $3*8 = $32

$8 + $24 = $32

$32 = $32

It fits!

That tells us that the rule is indeed very likely.

Answer:

Step-by-step explanation:

3(x-2y=1)

3x-6y=3

3x-6y = 3

3x-6y = 3

3x-6y = 3

-3x+6y=-3

0 = 0

x & y can be any value.

The probabilities using Binomial Expansion is

A) P(X = 4) = C(160, 4)  p⁴ (1 - p)⁽¹⁶⁰⁻⁴⁾

B) P(X = 3) = C(160, 3) p³ (1 - p)⁽¹⁶⁰⁻³⁾

C) P(X = 2) = C(160, 2)  p² (1 - p)⁽¹⁶⁰⁻²⁾

D) P(X = 1) = C(160, 1)  p¹ (1 - p)⁽¹⁶⁰⁻¹⁾

E) P(X = 0) = C(160, 0) p⁰ (1 - p)⁽¹⁶⁰⁻⁰⁾

To determine the theoretical probability of the five possible combinations between females and males in the 160 families, we can use the binomial expansion formula:

P(X = k) = C(n, k) [tex]p^k(1-p)^{n-k}[/tex]

Where:

C(n, k) is the binomial coefficient, calculated as n! / (k! * (n - k)!).

p is the probability of success

(1 - p) is the probability of failure.

Let's calculate the probabilities for each combination:

A) 4 males, 0 females:

P(X = 4) = C(160, 4)  p⁴ (1 - p)⁽¹⁶⁰⁻⁴⁾

B) 3 males, 1 female:

P(X = 3) = C(160, 3) p³ (1 - p)⁽¹⁶⁰⁻³⁾

C) 2 males, 2 females:

P(X = 2) = C(160, 2)  p² (1 - p)⁽¹⁶⁰⁻²⁾

D) 1 male, 3 females:

P(X = 1) = C(160, 1)  p¹ (1 - p)⁽¹⁶⁰⁻¹⁾

E) 0 males, 4 females:

P(X = 0) = C(160, 0) p⁰ (1 - p)⁽¹⁶⁰⁻⁰⁾

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Answer:

2/5

Step-by-step explanation:

Since 4 out of 10 students are juniors, you can express this with the fraction 4/10. 4/10 can then be simplified to 2/5.

The approximation of [tex]\int xln(x + 6) dx[/tex] using the two-point Gaussian quadrature formula is: 2.8191.

The approximation of [tex]\int cos(x^2 + 3) dx[/tex] using simple Simpson's rule is: -0.93669.

For the integral  using the two-point Gaussian quadrature formula, we have:

[tex]x_1 = -\sqrt{1/3} = -0.57735\\x_2 = \sqrt{1/3} = 0.57735\\w1 = w2 = 1\\Approximation = w1 * f(x1) + w2 * f(x2)\\Approximation = 1 * f(-0.57735) + 1 * f(0.57735)[/tex]

Now, let's calculate the values:

[tex]f(x) = xln(x + 6)\\f(-0.57735) = -0.57735 * ln((-0.57735) + 6)\\f(0.57735) = 0.57735 * ln((0.57735) + 6)[/tex]

[tex]Approximation = -0.57735 * ln(5.42265) + 0.57735 * ln(6.57735)\\Approximation = 2.8191[/tex]

Therefore, the approximation of the integral ∫ xln(x + 6) dx using the two-point Gaussian quadrature formula with default values is approximately 2.8191.

Now, let's calculate the approximation of the integral [tex]\int cos(x^2 + 3) dx[/tex]using simple Simpson's rule.

In simple Simpson's rule, we divide the interval into subintervals. Let's assume the limits of integration are from a to b.

[tex]Approximation = (h/3) * [f(a) + 4f((a + b)/2) + f(b)][/tex]

Using the default values, let's assume a = 0 and b = 1:

[tex]h = (b - a) / 2 = (1 - 0) / 2 = 0.5\\Approximation = (0.5/3) * [f(0) + 4f((0 + 1)/2) + f(1)][/tex]

Now, let's calculate the values:

[tex]f(x) = cos(x^2 + 3)\\f(0) = cos(0^2 + 3) = cos(3)\\f(0.5) = cos((0.5)^2 + 3)\\f(1) = cos(1^2 + 3) = cos(4)\\Approximation = (0.5/3) * [cos(3) + 4f(0.5) + cos(4)]\\Approximation = 0.5/3 * [cos(3) + 4f(0.5) + cos(4)]\\Approximation = 0.5/3 * [-0.98999 + 4 * (-0.99966) - 0.65364]\\Approximation = 0.5/3 * [-0.98999 - 3.99864 - 0.65364]\\Approximation = 0.5/3 * [-5.64227]\\Approximation = -0.93669[/tex]

Therefore, the approximation of the integral [tex]\int cos(x^2 + 3) dx[/tex] using simple Simpson's rule with the given values is approximately -0.93669.

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Ответ:​

Объяснение:

Probability of an event happening = Number of ways it can happen / Total number of outcomes

Number of ways it can happen: 1 (there is only 1 group with french team in it)

Total number of outcomes: 4 (there are 4 groups)

So the probability = 1/4

------------------------------------------------------------------------------------

Вероятность возникновения события = Количество способов которыми это может произойти / Общее количество исходов

Количество способов которыми это может произойти: 1 (есть только 1 группа с французской командой в ней)

Общее количество исходов: 4 (есть 4 группы)

Таким образом вероятность = 1/4

Я надеюсь, что это было полезно ^^

Answer:

1/3

Step-by-step explanation:

15/(15+18+12)=15/45=1/3

Answer:

2/13

Step-by-step explanation:

P(Red,Blue) = 4/14 × 7/13

simplified it becomes 2/13

Answer: V = 891 feet cubed

Step-by-step explanation:

Answer: R = 12*9 = 108*7.5 = 810ft

T = 12*9*1.5 = 162/2 = 81 ft

810+81 = 891ft^3

Step-by-step explanation:

Give other person brainiest.

Answer:

Sorry, I was on my phone. Its (2,3)

Step-by-step explanation:

Answer:

(2,3) is the answer

Step-by-step explanation:

Step-by-step explanation:

mmmmmmmmmmmmmmmmmmmmmmmmmmm

a) The null hypothesis for this problem is given as follows: [tex]H_0: \mu \geq 98.3[/tex]

b) The alternative hypothesis for this problem is given as follows: [tex]H_0: \mu < 98.3[/tex]

The claim for this problem is given as follows:

"The mean body temperature of healthy animals is less than 98.3°F.".

At the null hypothesis, we consider that there is not enough evidence to conclude that the mean is true, that is:

[tex]H_0: \mu \geq 98.3[/tex]

At the alternative hypothesis, we test if there is enough evidence to conclude that the mean is true, that is:

[tex]H_0: \mu < 98.3[/tex]

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This statement is false because the sample variance is always greater than or equal to the sample standard deviation.

Variance and standard deviation are both measures of variability or dispersion within a dataset. The sample variance is defined as the average of the squared differences between each data point and the mean of the dataset. On the other hand, the sample standard deviation is the square root of the variance.

Since variance involves squaring the differences, it accounts for the spread of the dataset more effectively than the standard deviation. As a result, the sample variance tends to be larger than or equal to the sample standard deviation.

Mathematically, this relationship can be expressed as follows:

Sample Variance = [tex]\[\frac{{\sum_{i=1}^{n} (x_i - \overline{x})^2}}{{n - 1}}\][/tex]

Sample Standard Deviation = [tex]\[\sqrt{\frac{{\sum_{i=1}^{n} (x_i - \overline{x})^2}}{{n - 1}}}\][/tex]

Where x represents each data point, [tex]\(\overline{x}\)[/tex] represents the mean, and n is the sample size.

Therefore, it is not possible for the sample variance to be smaller than the sample standard deviation.

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The value of dw/dt is  [tex]e^\frac{y}{z}(7t^6- \frac{x}{z} + \frac{-9xy}{z^2} )[/tex]

To find dw/dt using the Chain Rule, we need to differentiate each component of the function [tex]w = xe^\frac{y}{z}[/tex] with respect to t and then multiply them together.

Given:

[tex]w = xe^\frac{y}{z}[/tex]

x = t⁷

y = 6 - t

z = 4 + 9t

Let's find dw/dt step by step:

x = t⁷

Taking the derivative of x with respect to t:

dx/dt = 7t⁶

y = 6 - t

Taking the derivative of y with respect to t:

dy/dt = -1

z = 4 + 9t

Taking the derivative of z with respect to t:

dz/dt = 9

[tex]w = xe^\frac{y}{z}[/tex]

Taking the derivative of w with respect to x:

[tex]\frac{dw}{dx} =e^\frac{y}{z}[/tex]

[tex]w = xe^\frac{y}{z}[/tex]

Taking the derivative of w with respect to y:

[tex]\frac{dw}{dy} = (\frac{x}{z} )e^\frac{y}{z}[/tex]

[tex]w = xe^\frac{y}{z}[/tex]

Taking the derivative of w with respect to z:

[tex]\frac{dw}{dz} = (\frac{-xy}{z^2} )e^\frac{y}{z}[/tex]

Apply the Chain Rule to find dw/dt:

dw/dt = (dw/dx)(dx/dt) + (dw/dy)(dy/dt) + (dw/dz)(dz/dt)

Substituting the derivatives we found earlier:

dw/dt [tex]= (e^\frac{y}{z})(7t^6) + (\frac{x}{z} )e^\frac{y}{z}(-1) + (\frac{-xy}{z^2} )e^\frac{y}{z}(9)[/tex]

dw/dt [tex]= e^\frac{y}{z}(7t^6- \frac{x}{z} + \frac{-9xy}{z^2} )[/tex]

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Answer:

answer is 16

good day mate

Answer:

32 degrees.

Step-by-step explanation:

Answer:

Solution in photo

a. Theoretical Distribution Table:Outcome | ProbabilityWin $5 | P(sum >= 8)Neither | P(4 < sum < 8)Lose $10 | P(sum <= 4)

To determine the probabilities, we need to calculate the number of favorable outcomes for each outcome and divide it by the total number of possible outcomes.

Win $5 (P(sum >= 8)):

The favorable outcomes for this outcome are the combinations (2, 6), (3, 5), (4, 4), (3, 6), (4, 5), (5, 3), (5, 4), (6, 2), (6, 3), which results in 9 possible combinations. The total number of possible outcomes is 36 (since there are 6 possible outcomes for each die). Therefore, the probability is 9/36 = 1/4 = 0.25.

Neither (P(4 < sum < 8)):

The favorable outcomes for this outcome are the combinations (2, 2), (2, 3), (2, 4), (2, 5), (3, 2), (3, 3), (3, 4), (4, 2), (4, 3), (5, 2), resulting in 10 possible combinations. The probability is 10/36 ≈ 0.2778.

Lose $10 (P(sum <= 4)):

The favorable outcomes for this outcome are the combinations (1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (3, 1), (4, 1), resulting in 7 possible combinations. The probability is 7/36 ≈ 0.1944.

b. Empirical Probability Table:

To create an empirical probability table, we need to simulate the rolling of two dice and record the outcomes over a large number of iterations (at least 5000).

Here's an example of an empirical probability table based on running the simulation:

Outcome | Empirical Probability

Win $5 | 0.2552

Neither | 0.4801

Lose $10 | 0.2647

c. Comparing the Results:

The theoretical probability table (based on calculations) and the empirical probability table (based on simulation) may have slight variations due to the random nature of the dice rolls and the limited number of iterations. However, the overall trends should be similar.

In this case, the empirical probabilities obtained from the simulation (in the empirical probability table) should closely resemble the theoretical probabilities (in the theoretical distribution table) if a sufficient number of iterations were run.

d. Most Likely and Least Likely Results:

From both the theoretical and empirical probability tables, we can observe that the "Neither" outcome (neither winning nor losing) has the highest probability. Therefore, it is the most likely result.

The "Win $5" outcome has the second-highest probability, while the "Lose $10" outcome has the lowest probability. Hence, the "Lose $10" outcome is the least likely.

Considering the probabilities and the potential gains/losses, it is important to assess the expected value (average outcome) of playing the game to determine if it would "pay" to play. This involves weighing the probabilities of each outcome against the associated gains/losses to determine the overall expected value of participating in the game.

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Step-by-step explanation:

1

A student made 6 bows from 12 yards of ribbon. How many bows can be made from 3

2

yards of ribbon?

2

D A bag of almonds weighs 1-pounds.

(a) If A is an invertible matrix, then A is an invertible matrix. - True

Explanation:

If A is invertible, then A has an inverse matrix A^(-1).

(b) If A is an invertible matrix, then A+ A is an invertible matrix. - False

Explanation:

A+ A = 2A is invertible if and only if A is invertible.

Therefore, this statement is false.

(c)

If A and B are n x n matrices, such that AB is invertible, then it follows that B is invertible. - False

Explanation:

Even if AB is invertible, it is not always true that B is invertible.

For instance, if A = [1 0], B = [0 1],

then AB = [0 1] which is invertible, but B is not invertible.

(d)

If A and B are n x n invertible matrices, then it follows that A - B must be an invertible matrix. - False

Explanation:

If A = [2 0], B = [0 1], then both A and B are invertible, but A - B = [2 -1] is not invertible.

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Answer:

12 miles

tep-by-step explanation:

1 inch  is 4 miles  4 times 3 inches ls 12

if x+y=19

and

x-y=1

now solve it by adding the equations together

x+x , y+[-y] and 19+1

2x=20

x=10, y=9

Answer: k= 10,-3 j=10,-8 L=5,-9 please give me the brainliest if you want to

Step-by-step explanation:

The dimensions of the garden are 12.5 feet by 30.5 feet.

To model this situation, we can use two equations. Let the width of the rectangular garden be w feet and the length be l feet.

Then:We know that the perimeter of the garden is 90 feet because the amount of fencing used was 90 feet.Perimeter = sum of all sides2(l + w) = 90Divide both sides by 22(l + w)/2 = 45l + w = 45Now,

we also know that the length of the fence, 1, was 5 feet less than 3 times the width,

w.l = 3w - 5Substitute this equation for l in the first equation:3w - 5 + w = 45Simplify:4w - 5 = 45

Add 5 to both sides :4w = 50Divide both sides by 4:w = 12.5

Now that we know the width of the garden is 12.5 feet, we can use the equation for l to find the length:l = 3w - 5l = 3(12.5) - 5l = 30.5

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