There are six nickels and six dimes in your
pocket. You randomly pick a coin out of
your pocket and place it on a counter.
Then you randomly pick another coin.
Both coins are nickels.

Using the given scale, we can see that 16 feet are represented by 4 inches.

Here we need to use the given scale.

We know the relation:

(1/4) inches = 1 feet

If we multiply both sides by 4, then we will get:

4*(1/4) inches = 4*1 feet

1 inch = 4 feet

Now we can do the change of units:

4 inches = 4*( 1 inch) = 4*4 feet = 16 feet

Then we can see that 4 inches is equal to 16 feet when using the given scale.

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To find ∂2z/(∂r ∂s), we need to take the partial derivative of ∂z/∂s with respect to r or the partial derivative of ∂z/∂r with respect to s.

Let's start by finding the partial derivatives of z with respect to x and y:

∂z/∂x = ∂f/∂x = 2r s2 ∂f/∂u    (where u = x = r2 s2)

∂z/∂y = ∂f/∂y = 9r ∂f/∂v    (where v = y = 9rs)

Next, we can use the chain rule to find the second partial derivative of z with respect to r and s:

∂2z/(∂r ∂s) = ∂/∂r (∂z/∂s)

= ∂/∂r (9s ∂f/∂v)          (since ∂z/∂s = ∂f/∂y = 9r ∂f/∂v)

= 9 ∂/∂r (s ∂f/∂v)

= 9 (∂/∂v (s ∂f/∂u) * ∂u/∂r + ∂/∂v (s ∂f/∂v) * ∂v/∂r)

= 9 (s ∂2f/∂u∂v * 2rs + ∂f/∂v * 9)

= 18rs s ∂2f/∂u∂v + 81r ∂f/∂v

Therefore, the expression for ∂2z/(∂r ∂s) in terms of f(x,y) is:

∂2z/(∂r ∂s) = 18rs s ∂2f/∂u∂v + 81r ∂f/∂v, where u = x = r2 s2 and v = y = 9rs.

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a. On average, callers must wait 17.14 minutes before talking to the sales rep.

b. On average, there are 12.73 customers on hold.

c. The probability that a customer will be placed on hold is 0.9637 or 96.37%.

d. The sales rep's utilization rate is 95.24%.

e. Road Rambler should employ two sales reps to ensure no more than a 10% chance of customers being placed on hold.

a. The average time a caller must wait before talking to the sales rep is the sum of the average time between calls (4.29 minutes) and the average time it takes to service a call (4 minutes), which equals 8.29 minutes. Thus, the total average waiting time is 14.57 minutes per customer, resulting in 17.14 minutes of waiting time for each caller.

b. The average number of customers on hold is equal to the average time a customer spends waiting divided by the average time between calls, which is:

= 14.57/4.29

= 3.39

Therefore, the average number of customers on hold is 3.39 x 14 = 12.73.

c. The probability that a customer will be placed on hold is equal to the probability that there are more than zero customers in the system. Using the formula for the probability of zero customers in the system (0.0363), we can calculate that the probability of at least one customer in the system is:

= 1 - 0.0363

= 0.9637 or 96.37%.

d. The sales rep's utilization rate is the ratio of the average service time to the average time between calls, which is:

= 4/4.29

= 0.9524 or 95.24%.

e. To ensure that no more than a 10% chance of customers being placed on hold, the probability of zero customers in the system must be at least 0.1. Using the formula for the probability of zero customers in the system, we can solve for the required arrival rate:

λ = -ln(0.9)/4.29

λ = 0.1822 customers per minute.

This arrival rate requires two sales reps, assuming no idle time, to handle the calls.

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Step-by-step explanation:

devide 21 by 7.5 and then multiply by 100

Answer:

35

Step-by-step explanation:

(-18) + (-83) +142) + |15) + (-21)

41 + 15 - 21

56 - 21

35

The standard form of the equation of the hyperbola is ((x - 6)^2 / 9) - ((y - 0)^2 / 27) = 1 or equivalently ((x - 6)^2 / (3^2)) - ((y - 0)^2 / (3sqrt(3))^2) = 1.

Since the foci of the hyperbola lie on the x-axis, we know that the transverse axis is horizontal. The center of the hyperbola is the midpoint between the vertices, which is ((3+9)/2, 0) = (6, 0). The distance between the center and each vertex is a = (9-3)/2 = 3, and the distance between the center and each focus is c = 12/2 = 6. The distance between each focus and vertex is b, where b^2 = c^2 - a^2 = 36 - 9 = 27, so b = sqrt(27) = 3sqrt(3).

Therefore, the standard form of the equation of the hyperbola is:

((x - 6)^2 / 9) - ((y - 0)^2 / 27) = 1

or equivalently:

((x - 6)^2 / (3^2)) - ((y - 0)^2 / (3sqrt(3))^2) = 1

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The expression is s2 = {(b, a), (b, d), (d, c), (d, d)}.

To find s2, which is the composition of the relation s with itself, we need to apply each element of s to itself and then combine the results.

The elements of s2 are of the form (x, z), where there exists some y such that (x, y) ∈ s and (y, z) ∈ s.

Using the given relation s = {(a,c),(b,d),(d,a)}, we can find s2 as follows:

(a, c) ∈ s, (c, a) ∉ s, so (a, a) ∉ s2

(a, c) ∈ s, (c, b) ∉ s, so (a, b) ∉ s2

(a, c) ∈ s, (c, d) ∉ s, so (a, d) ∉ s2

(a, c) ∈ s, (c, a) ∉ s, so (a, a) ∉ s2

(b, d) ∈ s, (d, a) ∈ s, so (b, a) ∈ s2

(b, d) ∈ s, (d, b) ∉ s, so (b, b) ∉ s2

(b, d) ∈ s, (d, d) ∈ s, so (b, d) ∈ s2

(b, d) ∈ s, (d, a) ∈ s, so (b, a) ∈ s2

(d, a) ∈ s, (a, c) ∈ s, so (d, c) ∈ s2

(d, a) ∈ s, (a, d) ∈ s, so (d, d) ∈ s2

(d, a) ∈ s, (a, a) ∉ s, so (d, a) ∉ s2

(d, a) ∈ s, (a, c) ∈ s, so (d, c) ∈ s2

Therefore, s2 = {(b, a), (b, d), (d, c), (d, d)}.

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The value of x is equal to 7.5 units.

In order to determine the value of x, we would apply cosine ratio because the given side lengths represent the adjacent side and hypotenuse of a right-angled triangle.

cos(θ) = Adj/Hyp

Where:

By substituting the given side lengths cosine ratio formula, we have the following;

cos(θ) = Adj/Hyp

cos(20) = x/8

x = 8cos(20)

x = 7.5 units.

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The graph of the function f(t) = t(1 − u1(t)) + et(u1(t) − u2(t)) consists of a straight line with slope 1 for t < 0, a decreasing curve for 0 ≤ t < 1, and a horizontal line at y = 0 for t ≥ 1. The curve starts at (0,0) and approaches the t-axis asymptotically as t increases for 0 ≤ t < 1.

The function f(t) can be written as a piecewise function:

f(t) = { t, if t < 0,

t-e^(-t), if 0 ≤ t < 1,

0, if t ≥ 1.

To sketch the graph of f(t), we can first plot the graph of each piece of the function and then combine them.

For t < 0, the graph of f(t) is just the line y = t. For 0 ≤ t < 1, the graph of f(t) is the line y = t minus the decreasing exponential curve y = -e^(-t), which intersects the t-axis at (0,-1) and approaches the t-axis asymptotically as t increases. For t ≥ 1, the graph of f(t) is the horizontal line y = 0.

Combining these pieces, we get the following graph of f(t):

          |

       ---|---

          |

          |

          |

----------|-----------

          |

          |

          |

          |

The graph consists of a straight line with slope 1 for t < 0, a decreasing curve that starts at (0,0) and approaches the t-axis asymptotically as t increases, for 0 ≤ t < 1, and a horizontal line at y = 0 for t ≥ 1.

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Answer:

31x²

Step-by-step explanation:

Area of triangle = 0.5 X 5 X 2 = 5  (5x square units)

Area of square = 6x X 6x = 36x square units

36x² - 5x² = 31x²

The exact length of the curve is:

[tex]L = 2(17^\frac{2}{3}-1 )[/tex]

We have the values of x and y are:

x = 6 + 3[tex]t^2[/tex]___eq.(1)

y = 8 + 2[tex]t^3[/tex]___eq.(2)

We have to find the exact length of the curve.

Now, According to the question:

We have to use the formula for length L of the curve:

[tex]L=\int\limits^4_0 \sqrt{[x'(t)]^2+[y'(t)]^2} \, dt[/tex]

Now, Differentiate both equations:

x' = 6t

y' = [tex]6t^2[/tex]

Plug the values in above formula:

[tex]L=\int\limits^4_0 \sqrt{6^2t^2+6^2t^4} \, dt[/tex]

By pulling 6t out of the square-root,

[tex]L=\int\limits^4_06t {\sqrt{1+t^2} } \, dt[/tex]

by rewriting a bit further,

[tex]L=3\int\limits^4_02t ({1+t^2} )^\frac{1}{2} \, dt[/tex]

by General Power Rule,

[tex]L = 3[\frac{2}{3}(1+t^2)^\frac{3}{2} ]^4_0[/tex]

[tex]L = 2(17^\frac{2}{3}-1 )[/tex]

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Therefore, ∂³w/∂z∂y∂x = 2y using indicated partial derivatives.

To find the indicated partial derivative, we start by computing the partial derivatives of w with respect to x, y, and z.

∂w/∂x = y * 2z = 2yz

∂w/∂y = x * 2y * 2z = 4xyz

∂w/∂z = x * y * 2 = 2xy

Now we can find the third-order mixed partial derivative by differentiating ∂w/∂z with respect to y and then with respect to x:

∂³w/∂z∂y∂x = ∂²/∂x∂y (2xy) = 2y

Therefore, ∂³w/∂z∂y∂x = 2y.

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a. The 90% confidence interval for the difference between the corresponding population means is (2.23, 10.77) hours of free time per week.

b. At a 5% significance level, the null hypothesis that the two population means are equal is rejected, suggesting that there is evidence to support the claim that tenth-graders have more free time than twelfth-graders.

To calculate the 90% confidence interval for the difference between the corresponding population means, we can use the formula:

(x1 -x2) ± tα/2,df * SE

where:

x1 = 29 hours (the mean free time of the tenth-graders)

x2 = 22 hours (the mean free time of the twelfth-graders)

tα/2,df is the t-value from the t-distribution table with α/2 = 0.05/2 = 0.025 and degrees of freedom df = n1 + n2 - 2 = 25 + 23 - 2 = 46

SE is the standard error of the difference between the sample means, calculated as:

SE =[tex]\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^{0.5}[/tex], where s1 = 7.0, s2 = 6.2, n1 = 25, and n2 = 23.

Substituting the values, we get:

(29 - 22) ± 2.063 * [tex]\left(\frac{{7.0^2}}{{25}} + \frac{{6.2^2}}{{23}}\right)^{0.5}[/tex]

= 7 ± 2.777

So the 90% confidence interval is (7 - 2.777, 7 + 2.777), or (2.23, 10.77).

This means that we are 90% confident that the true difference between the population means of the two groups falls within the range of 2.23 to 10.77 hours of free time per week.

b. To test the hypothesis that the two population means are different, we can use a two-sample t-test with the null hypothesis H0: μ1 - μ2 = 0 and the alternative hypothesis Ha: μ1 - μ2 ≠ 0, where μ1 and μ2 are the population means of the two groups.

The test statistic is calculated as:

t = (x1 - x2) / SE, where SE is the standard error of the difference between the sample means, calculated as:

SE =[tex]\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^{0.5}[/tex], where s1 = 7.0, s2 = 6.2, n1 = 25, and n2 = 23.

Substituting the values, we get:

SE = [tex]\left(\frac{7.0^2}{25} + \frac{6.2^2}{23}\right)^{0.5}[/tex] = 1.974

The test statistic is:

t = (29 - 22) / 1.974

t = 3.54

The complete question:

A high school counselor wanted to know if tenth-graders at her high school tend to have the same free time as the twelfth-graders. She took random samples of 25 tenth-graders and 23 twelfth-graders. Each student was asked to record the amount of free time he or she had in a typical week. The mean for the tenth-graders was found to be 29 hours of free time per week with a standard deviation of 7.0 hours. For the twelfth-graders, the mean was 22 hours of free time per week with a standard deviation of 6.2 hours. Assume that the two populations are approximately normally distributed with unknown but equal standard deviations.

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Therefore, the conditional probability that East has 3 spades given that North and South have a combined total of 8 spades is approximately 0.107.

To calculate the conditional probability that East has 3 spades given that North and South have a combined total of 8 spades, we need to use Bayes' Theorem.

Let A be the event that East has 3 spades and B be the event that North and South have a combined total of 8 spades. Then, we need to find P(A|B), which is the probability that East has 3 spades given that North and South have a combined total of 8 spades.

Bayes' Theorem states that:

P(A|B) = P(B|A) * P(A) / P(B)

where P(B|A) is the probability of North and South having a combined total of 8 spades given that East has 3 spades, P(A) is the probability of East having 3 spades, and P(B) is the probability of North and South having a combined total of 8 spades.

We don't have information to calculate these probabilities directly, so we need to use some assumptions. One common assumption is that each player has an equal chance of having any particular suit, and the distribution of suits is independent across players.

Under this assumption, the probability of East having 3 spades is the probability of drawing 3 spades from the remaining 10 spades, which is

P(A) = (10 choose 3) / (52 choose 13) ≈ 0.098

The probability of North and South having a combined total of 8 spades is the sum of the probabilities of the following cases:

North has 5 spades and South has 3 spades

North has 4 spades and South has 4 spades

North has 3 spades and South has 5 spades

Under our assumption, the probability of each of these cases is:

P(North has k spades) * P(South has 8-k spades) = (13 choose k) * (39 choose 8-k) / (52 choose 13)

Therefore,

P(B) = [ (13 choose 5) * (39 choose 3) + (13 choose 4) * (39 choose 4) + (13 choose 3) * (39 choose 5) ] / (52 choose 13) ≈ 0.211

Finally, we need to calculate P(B|A), which is the probability of North and South having a combined total of 8 spades given that East has 3 spades. Under our assumption, this probability can be calculated as:

P(B|A) = P(North and South have 5 spades in total) = (3 choose 2) * (10 choose 1) / (13 choose 2) ≈ 0.231

Putting it all together, we get:

P(A|B) ≈ P(B|A) * P(A) / P(B) ≈ 0.231 * 0.098 / 0.211 ≈ 0.107

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True.  w is a subspace of rn and if v is in both w and w⊥, then v must be the zero vector.

By definition, the orthogonal complement of a subspace W of R^n is the set of all vectors in R^n that are orthogonal to every vector in W. So, W ⊥ consists of all vectors that are orthogonal to every vector in W. If v is in both W and W ⊥, then v must be orthogonal to itself (since it's in W and W ⊥, it must be orthogonal to every vector in W, including itself). This means that v must be the zero vector, since the only vector that is orthogonal to itself is the zero vector. Therefore, the statement is true.

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In the given study, the correct symbols representing the underlined numbers are as follows:

- 6.8 hours is represented by μ (mu). The symbol μ is used to represent the population mean.

- 1.5 hours is represented by σ (sigma). The symbol σ is used to represent the population standard deviation.

- 30 NIACC students is represented by n. The lowercase letter n is commonly used to denote the sample size.

- < 6.4 hours is represented by[tex]x (x-bar)[/tex]. The symbol [tex]x bar[/tex] is used to represent the sample mean.

Therefore, it's important to note that the study is comparing the sleep habits of college students (population) to a sample of NIACC students. The population mean and standard deviation are given for the college students, while the sample mean is provided for the NIACC students.

The sample size (number of NIACC students) is also given. These symbols allow for the distinction between population parameters and sample statistics, which is crucial in statistical analysis and inference.

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Answer:

2/11 because A would be 0.5, B would be 0.2 and D would be 0.125. 2/11=0.18181818181818... making it a repeating decimal

Step-by-step explanation:

To express 7 1/5 + (-8 3/5) as the sum of its integer and fractional parts, first we'll calculate the sum by breaking it down into integers and fractions.

1. Add the integers together: 7 + (-8) = -1

2. Add the fractional parts together: 1/5 + (-3/5) = -2/5

Combined, the sum can be expressed as:

-1 2/5 (or -1 and -2/5 if you want to keep the sum in mixed number notation).

Answer:

The sum of the integer and fractional parts is -1 and -2/5, respectively. And can be represented as -7/5.

What is a fraction number?

A fraction number is a number that represents the part of the whole, where the whole can be any number. It is in the form of numerator and denominator.

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If on a turn, the player moves forward the number of spaces equal to the first number rolled and then moves backward the number of spaces equal to the second number rolled. Then the probability that a player ends up 2 spaces ahead of where he started on a turn is 11%.

To end up 2 spaces ahead of where the player started, the number of spaces forward should be more than the number of spaces backward by 2.

⇒ First number - Second number = 2

All satisfied outcomes = (3,1), (4,2), (5,3) and (6,4).

Total possible outcomes = 36(6 outcomes for the first roll multiplied by 6 outcomes for the second roll)

[tex]Probability = \frac{All \ satisfied \ outcomes}{Total \ possible \ outcomes} = \frac{4}{36} = \frac{1}{9}[/tex]

Converting this to percentage,

[tex]\frac{1}{9}[/tex] x 100 = 11.11 ≈ 11%

Therefore, the probability that a player ends up 2 spaces ahead of where he started on a turn is 11%.

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The equation of the curve that passes through the point (9,8) and has a slope of 3x at each point is:

y = (3/2)x²  - 94.5

To find the curve in the xy-plane that passes through the point (9,8) and whose slope at each point is 3x, we can use calculus to solve for the equation of the curve.

First, we integrate the given slope function with respect to x to obtain the expression for the curve's vertical position y:

dy/dx = 3x

dy = 3x dx

Integrating both sides:

y = (3/2)x²  + C (where C is the constant of integration)

Next, we can use the given point (9,8) to find the value of the constant C:

8 = (3/2)(9)²  + C

C = 8 - (3/2)(81) = -94.5

Therefore, the equation of the curve that passes through the point (9,8) and has a slope of 3x at each point is:

y = (3/2)x - 94.5

We can graph this equation to visualize the curve.

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To determine the size of Stephanie's monthly payments for the safe annuity, we need to calculate the annuity payment amount that will accumulate to $1,200,000 by the time she reaches age 60.

Given that the annuity is compounded monthly at a rate of 6.75%, we can use the formula for the future value of an ordinary annuity:

Future Value = Payment Amount × [(1 + r)^n - 1] / r,

where r is the monthly interest rate (6.75% divided by 100 and then by 12) and n is the number of compounding periods (number of months from age 20 to age 60, which is 40 years multiplied by 12 months per year).

By substituting the values into the formula, we can solve for the payment amount:

$1,200,000 = Payment Amount × [(1 + 0.0675/12)^(40×12) - 1] / (0.0675/12).

Simplifying the equation and solving for the payment amount, we find that Stephanie's monthly payments would need to be approximately $1,404.64 to accumulate $1,200,000 by the time she reaches age 60.

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Using the formula of length of an arc, the length of the arc of the given circle is approximately 2.2m

The length of an arc of a circle is the total length in which the arc makes with the circle.

The arc length of a circle with radius 2.54 meters and angle 50 degrees is 1.79 meters. This can be calculated using the following formula:

arc length = θ/360 * 2πr

Substituting in the values into the formula;

Length of arc = (50/360) * 2π * 2.54

Length of arc = 127π/180 m

Length of arc = 2.2m

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Translation and complete question: What is the arc length for an angle of 50 degree is a circle radius of 2.54 meters

To make the equation 4x - 4 = 4x + _____ true for all values of x, we can write "0" in the blank. This will result in an equation that is always true regardless of the value of x.

By examining the given equation, we can see that the left-hand side (4x - 4) and the right-hand side (4x + _____) have the same expression, which is 4x. To make the equation true for all values of x, we need the expressions on both sides to be equal.

By adding "0" to the right-hand side, the equation becomes 4x - 4 = 4x + 0. Since the two expressions on both sides are now identical (both equal to 4x), the equation holds true for all values of x.

Adding 0 to an expression does not change its value, so the equation 4x - 4 = 4x + 0 is satisfied for any value of x, making it true for all values of x.

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Brandon does not have enough flour to make 2 batches of bagels

An equation is an expression that is used to show how numbers and variables are related using mathematical operators

A recipe for one batch of bagels uses 1 3/4 cups of flour.

Amount of flour to make 2 batches of bagels = 1 3/4 + 1 3/4 = 7/4 + 7/4 = 7/2 = 3.5 cups of flour

Brandon has 3 1/8 cups of flour = 3.125 cups

1 3/4 + 1 3/4 (3.5 cups of flour) is greater than 3 1/8 (3.125 cups)

Brandon does not have enough flour to make 2 batches of bagels

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The probability that a randomly selected park with newly updated picnic tables is a rural park is 2/7 .The probability is approximately 0.286 or 28.6%.

To find the probability that a randomly selected park with newly updated picnic tables is a rural park, we need to use conditional probability. We know the number of urban parks (30) and rural parks (18), as well as the number of urban parks updated with new picnic tables (10) and rural parks updated with new picnic tables (4).

Let's define the events:

A: Park is rural

B: Park is updated with new picnic tables

We want to find P(A|B), which represents the probability that the park is rural given that it is updated with new picnic tables.

Using the formula for conditional probability:

P(A|B) = P(A ∩ B) / P(B)

P(A ∩ B) represents the probability of both events A and B occurring. In this case, it is the probability that a park is both rural and updated with new picnic tables. From the information given, we know that 4 rural parks were updated with new picnic tables, so P(A ∩ B) = 4.

P(B) represents the probability of event B occurring, which is the probability that a park is updated with new picnic tables. This is the sum of the urban and rural parks that were updated, which is 10 + 4 = 14.

Now we can calculate P(A|B):

P(A|B) = P(A ∩ B) / P(B) = 4 / 14 = 2/7

Therefore, the probability that a randomly selected park with newly updated picnic tables is a rural park is 2/7.

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The sequence converges to ln 2.

To determine whether the sequence converges or diverges, we need to take the limit as n approaches infinity. We can simplify the sequence by using the properties of logarithms:

an = ln( 2n^2 + 1 ) - ln( n^2 +1 )

= ln[(2n^2 + 1)/(n^2 + 1)]

Now we can take the limit of this expression as n approaches infinity:

lim [ln[(2n^2 + 1)/(n^2 + 1)]]

n→∞

Using L'Hopital's Rule, we can evaluate this limit:

Lim [(4n)/(2n)]

n→∞

This limit is equal to 2. Therefore, the sequence converges to ln 2.

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To find the slope of the tangent line and the instantaneous rate of change of the function at the point (-2,0), we first need to find the derivative of the function:

y = (x^2 + 3)(x^3 - 4x)

y' = [(2x)(x^3 - 4x) + (x^2 + 3)(3x^2 - 4)]

= 2x^4 - 8x^2 + 3x^2 - 4

= 2x^4 - 5x^2 - 4

(a) To find the slope of the tangent line at (-2,0), we substitute x = -2 into the derivative:

y' = 2(-2)^4 - 5(-2)^2 - 4 = 24

Therefore, the slope of the tangent line at (-2,0) is 24.

(b) The instantaneous rate of change of the function at (-2,0) is also given by the derivative at that point:

y'(-2) = 2(-2)^4 - 5(-2)^2 - 4 = 24

Therefore, the instantaneous rate of change of the function at (-2,0) is 24.

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the volume in the first octant bounded by[tex]y^2=4−x[/tex] and y=2z is pi/36 sqrt(3).

(1) To find the volume in the first octant bounded by the surfaces [tex]y^2 = 4 - x[/tex] and y = 2z, we can set up a triple integral in cylindrical coordinates.

First, we need to determine the bounds for our variables. Since we are working in the first octant, we know that 0 <= z, 0 <= theta <= pi/2, and 0 <= r.

Next, we need to find the equation for the upper and lower bounds of z in terms of r and theta. We can start with the equation [tex]y^2 = 4 - x[/tex] and substitute y = 2z to get:

[tex](2z)^2 = 4 - x[/tex]

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

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

We can then use this equation along with the equation z = y/2 to get the bounds for z:

[tex]0 < = z < = (4 - x)^(1/2)/2 = (4 - 4z^2)^(1/2)/2[/tex]

Squaring both sides, we get:

[tex]0 < = z^2 < = (1 - z^2)/2[/tex]

[tex]0 < = 2z^2 < = 1 - z^2[/tex]

[tex]z^2 < = 1/3[/tex]

So the bounds for z are:

[tex]0 < = z < = (1/3)^(1/2)[/tex]

Finally, we can set up the triple integral in cylindrical coordinates:

V = ∫∫∫ r dz dtheta dr

with bounds:

0 <= r

0 <= theta <= pi/2

[tex]0 < = z < = (1/3)^(1/2)[/tex]

and integrand:

r

So the volume in the first octant bounded by y^2=4−x and y=2z is:

V = ∫∫∫ r dz dtheta dr

= ∫ from 0 to[tex](1/3)^(1/2) ∫ from 0 to pi/2 ∫ from 0 to r r dz dtheta dr[/tex]

= ∫ from 0 to[tex](1/3)^(1/2) ∫ from 0 to pi/2 r^2/2 dtheta dr[/tex]

= ∫ from 0 to[tex](1/3)^(1/2) r^2 pi/4 dr[/tex]

[tex]= pi/12 (1/3)^(3/2)[/tex]

= pi/36 sqrt(3)

Therefore, the volume in the first octant bounded by[tex]y^2=4−x[/tex] and y=2z is pi/36 sqrt(3).

(2) To find the volume bounded by z = x^2 + y^2 and z = 4, we can use a triple integral in cylindrical coordinates.

First, we need to determine the bounds for our variables. Since we are working in the region where z is bounded by [tex]z = x^2 + y^2[/tex] and z = 4, we know that 0 <= z <= 4.

Next, we can rewrite the equation [tex]z = x^2 + y^2[/tex] in cylindrical coordinates as [tex]z = r^2.[/tex]

So the bounds for r and theta are:

0 <= r <= 2

0 <= theta <= 2pi

And the bounds for z are:

[tex]r^2 < = z < = 4[/tex]

Finally, we can set up the triple integral in cylindrical coordinates:

V = ∫∫∫ r dz dtheta dr

with bounds:

0 <= r <= 2

0 <= theta <= 2pi

[tex]r^2 < = z < = 4[/tex]

and integrand: 1

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The speed s of Ed's drive to work is 24 mph.

Let d be the distance between Ed's home and work, and let t be the time it takes him to drive to work at speed s. Then, we know that Ed drove half the distance to work before turning around, so he drove d/2 miles.

When he turns around, he increases his speed by 8 mph, so his new speed is s+8 mph. He then drives back to his home, covering the same distance of d/2 miles at an increased speed.

The time it takes him to drive home is therefore (d/2)/(s+8) hours. When he arrives home, he turns around and drives to work at a speed of (s+6) mph. The time it takes him to drive to work is then d/(s+6) hours.

Since Ed's total driving time is 67% greater than usual, we know that his new driving time is 1.67t. Setting up an equation using these values, we get:

d/(s+6) + d/2/(s+8) + d/(s) = 1.67t

Simplifying this equation yields:

d = 6t(s+6)/(7s+24)

We also know that Ed drove half the distance to before turning around, so:

d/2 = st/2

Substituting for d and solving for s yields:

s = 24 mph

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