Find an equation of the tangent to the curve at the given point by both eliminating the parameter and without eliminating the parameter. x = 4 + in t, y = t^2 + 6, (4, 7) y =

Using the definition of the integral given in the theorem, the value of the integral 5 to 1 of (x² - 4x + 8) dx is found to be equal to approximately 83.33.

The integral can be evaluated using the fundamental theorem of calculus, which states that the definite integral of a function can be found by evaluating its antiderivative at the limits of integration.

The antiderivative of (x² − 4x + 8) is (1/3)x³ - 2x² + 8x, so evaluating at the limits of integration 5 and 1 gives

(1/3)(5³) - 2(5²) + 8(5) - [(1/3)(1³) - 2(1²) + 8(1)]

= (125/3) - 50 + 40 - (1/3) + 2 - 8

= 83.33

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To show that v + w and v - w are orthogonal, we need to prove that their dot product is equal to zero. We have shown that if ||v|| = ||w|| and ||v - w|| = 0, then v + w and v - w are orthogonal.

First, let's express v and w in terms of their magnitudes and directions:
v = ||v||u
w = ||w||u'
where u and u' are unit vectors in the direction of v and w, respectively.
Then, we can write:
v + w = ||v||u + ||w||u'
v - w = ||v||u - ||w||u'
Now, let's take the dot product of v + w and v - w:
(v + w) · (v - w) = ||v||^2u · u - ||w||^2u' · u'
Note that u · u' = cos θ, where θ is the angle between u and u'. Since ||v|| and ||w|| are positive, we have:
||v||^2u · u - ||w||^2u' · u' = ||v||^2cos θ - ||w||^2cos θ
= (||v||^2 - ||w||^2)cos θ
But we know that ||v|| = ||w||, since ||v - w|| = 0. Therefore:
(||v||^2 - ||w||^2)cos θ = 0
Since cos θ ≠ 0 (otherwise u and u' would be orthogonal), we must have:
(||v||^2 - ||w||^2) = 0
which implies that ||v|| = ||w||.

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Answer: 1/50

Step-by-step explanation:

Step 1: We need to multiply the numerator and denominator by 100 since there are 2 digits after the decimal.

0.02 = (0.02 × 100) / 100

= 2 / 100 [ since 0.02 × 100 = 2 ]

Step 2: Reduce the obtained fraction to the lowest term

Since 2 is the common factor of 2 and 100 so we divide both the numerator and denominator by 2.

2/100 = (2 ÷ 2) / (100 ÷ 2) = 1/50

The slant height of a pyramid is the height of the pyramid from the base up to the top of the pyramid, measured perpendicular to the base. To find the slant height of a pyramid, we need to know the base and the height of the pyramid.

In this case, the base of the pyramid is a square with side lengths of 30 inches. The height of the pyramid is 50 inches. To find the slant height, we can use the formula:

slant height = (height / 2) / tan(π/4)

where π is approximately equal to 3.14159.

Substituting the given values into the formula, we get:

slant height = (50 / 2) / tan(π/4)

= 25 / tan(π/4)

= 25 / 0.7853981633974483

≈ 32.85 inches

Therefore, the slant height of the pyramid is approximately 32.85 inches

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A falcon can travel 29,998 centimeters farther than a worm in 10 seconds.

Hal learns that a falcon travels about 0.3 kilometers in 10 seconds, and a worm travels about 2 centimeters in 10 seconds. To determine how much farther a falcon can travel than a worm in 10 seconds, we need to convert the distance traveled by the falcon from kilometers to centimeters.1 kilometer = 100,000 centimeters. So, 0.3 kilometers = 0.3 x 100,000 = 30,000 centimeters. Therefore, a falcon travels 30,000 centimeters in 10 seconds .A worm travels 2 centimeters in 10 seconds. To find out how much farther the falcon travels than the worm in 10 seconds, we need to subtract the distance the worm travels from the distance the falcon travels.30,000 - 2 = 29,998

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The rental charge, denoted as "y," is determined based on the duration of time, denoted as "x," for which the item or service is rented. Factors such as costs, demand, competition, and desired profit margins influence the specific pricing structure.

The rental charge, denoted as "y," is determined based on the amount of time, denoted as "x," that the item or service is rented for. The longer the duration of rental, the higher the rental charge tends to be. The specific pricing structure for rental charges varies depending on the industry, location, and specific rental service being provided.

Rental charges are typically set by the rental company or service provider and can be influenced by several factors. These factors may include the cost of acquiring and maintaining the rental item, overhead expenses such as storage or transportation costs, demand and market conditions, competition, and desired profit margins.

For example, in the context of car rentals, the rental charge may be based on a fixed rate per hour or may involve different rates for specific time increments (e.g., hourly, daily, weekly). Additionally, there may be additional fees or surcharges based on factors such as mileage, fuel usage, insurance coverage, or any optional extras chosen by the customer.

It's important to note that rental charges can vary significantly across different industries and types of rental services. For instance, the rental charges for equipment rentals, housing rentals, or event space rentals may have different pricing structures and factors influencing the overall cost.

Ultimately, the rental charge is determined by considering various factors that contribute to the cost of providing the rental service and the duration of time for which the item or service is rented.

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To find the length of the diameter of a circle, first rewrite the equation in the standard form of a circle equation, which is (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius.

To rewrite the given equation, we complete the square for both the x and y terms.

Starting with 3x² - 7x + 3y² - 6y - 3 = 0, we group the x and y terms separately and complete the square:

3x² - 7x + 3y² - 6y - 3 = (3x² - 7x) + (3y² - 6y) - 3 = 3(x² - (7/3)x) + 3(y² - 2y) - 3.

To complete the square, we need to add the square of half the coefficient of x and y, respectively, to both sides of the equation:

3(x² - (7/3)x + (7/6)²) + 3(y² - 2y + 1²) - 3 = 3(x - 7/6)² + 3(y - 1)² - 3 + 3(49/36) + 3 = 3(x - 7/6)² + 3(y - 1)² + 24/36.

Simplifying further, we have:

3(x - 7/6)² + 3(y - 1)² = 1.

Comparing this equation with the standard form (x - h)² + (y - k)² = r², we can see that the center of the circle is (7/6, 1) and the radius is √(1/3) = 1/√3.

The diameter of a circle is twice the radius, so the length of the diameter is 2 * (1/√3) = 2/√3 * (√3/√3) = 2√3/3.

Therefore, the length of the diameter of the circle is 2√3/3.

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The equivalent ratio table can be expressed as  

table 1;

The arrangement will be 7, 21 , 35 , 63

                                          3 , 9 , 15 , 27

Table 2;

The arrangement will be 5 ,10 , 25, 35

                                          9, 18, 27 , 63

Table 3;

The arrangement will be 10 , 20, 50 , 70

                                          13 , 26, 65, 91

Table 4;

The arrangement will be 11 , 22 ,44 , 88

                                          2 , 4  , 8 ,  16

From the table 1 we will need to multiply the first term of the first role and the second role by 3, 5 9 to complete the role.

From the table 2 we will need to multiply the first term of the first role and the second role by 2, 5 , 7 to complete the role.

From the table 3 we will need to multiply the first term of the first role and the second role by 2, 5, 7 to complete the role.

From the table4 we will need to multiply the first term of the first role and the second role by 2 , 4 , 8 to complete the role.

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Answer:

58.03 ft

Step-by-step explanation:

To solve for the total circumference of circle F, we can create a ratio of section angle measure to circumference. We know that these two attributes of a circle have a linear relationship because the formula for arc length ([tex]S = 2\pi r \cdot \frac{\theta}{360\°}[/tex]) relies proportionately on the radius and angle measure of the section.

angle measure : circumference

               290°   :   46.75 ft

We can multiply this ratio by [tex]\frac{360}{290}[/tex] to get the corresponding circumference for a 360° section (which is the entire circle).

[tex]\frac{360}{290}(290\° : 46.75 \text{ ft})[/tex]

[tex]= 360\° : \boxed{58.03 \text{ ft}}[/tex]

Therefore, the circumference of circle F is approximately 58.03 ft.

The probability of a Type I error is determined by the chosen significance level (α) and does not change based on the actual value being less than a specified threshold.

The probability of committing a Type I error is denoted by α (alpha), also known as the significance level. A Type I error occurs when you reject a null hypothesis when it is actually true. The value of α is set before conducting a hypothesis test and is typically set at 0.05 or 0.01, depending on the desired level of confidence.
In your question, it seems there might be some missing information. The symbol "=" and "100" are unclear, and the term "0" seems incomplete. However, I can provide a general idea about the probability of a Type I error when the actual value is less than a specified threshold.
When the actual value is less than the specified threshold, it means the null hypothesis is true. In this case, the probability of committing a Type I error remains the same as the predetermined significance level (α). This is because the probability of a Type I error is defined as the likelihood of rejecting a true null hypothesis, and it does not depend on the specific values of the test statistic.
In summary, the probability of a Type I error is determined by the chosen significance level (α) and does not change based on the actual value being less than a specified threshold.

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There are 5,040 different seating arrangements possible.

(a) To find the number of different juries possible, we can use the combination formula. We want to choose 6 men out of 14 and 6 women out of 13, so we have:

C(14, 6) x C(13, 6) = 1,352,697,600

Therefore, there are 1,352,697,600 different juries possible.

(b) To find the number of different seating arrangements possible, we can use the permutation formula. We know that we need to seat the jurors so that no two people of the same sex are seated next to each other. Let's start with the men - we have 6 men to seat, and they cannot be seated next to each other. We can think of this as creating "gaps" for the men to sit in. For example, if we have 6 men, we would need 7 gaps: _ M _ M _ M _ M _ M _ (where the underscores represent the gaps). Then we can choose which gaps the men will sit in, which we can do using the combination formula. We have 7 gaps to choose from, and we need to choose 6 of them for the men to sit in. Therefore, we have:

C(7, 6) = 7

Now we can seat the women in the gaps between the men. We have 6 women to seat, and we have 7 gaps for them to sit in (including the gaps at the ends). We can think of this as arranging the women and gaps in a line:

_ M _ M _ M _ M _ M _

We need to choose which 6 of the 7 gaps the women will sit in, and then arrange the women in those gaps. We can choose the gaps using the combination formula, and then arrange the women in those gaps using the permutation formula. Therefore, we have:

C(7, 6) x P(6, 6) = 7 x 720 = 5,040

Therefore, there are 5,040 different seating arrangements possible.

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The difference in interest earned per year when depositing x dollars in account B instead of account A is 0.005x dollars.

To calculate the difference in interest earned per year between account B and account A, we need to consider the interest rates of both accounts and the initial deposit amount.

Let's assume the initial deposit amount is x dollars.

For account A, with a simple annual interest rate of 3%, the interest earned per year can be calculated as:

Interest_A = (3/100) * x = 0.03x dollars

For account B, with a simple annual interest rate of 3.5%, the interest earned per year can be calculated as:Interest_B = (3.5/100) * x = 0.035x dollars

To find the difference in interest earned per year, we subtract the interest earned in account A from the interest earned in account B:

Difference = Interest_B - Interest_A = 0.035x - 0.03x = 0.005x dollars

Therefore, the difference in interest earned per year when depositing x dollars in account B instead of account A is 0.005x dollars.

This means that for each dollar deposited, account B earns an additional 0.005 dollars of interest compared to account A per year.

It's important to note that this calculation assumes simple interest and doesn't take into account compounding or any other fees or factors that may affect the actual interest earned.

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a. The appropriate hypothesis test for this scenario would be a 2-Sample t-Test, as we are comparing the average pounds of grapes on unfertilized vines to the average pounds of grapes on fertilized vines.

b. The appropriate hypothesis test for this scenario would be a 1-Sample t-Test, as we are comparing the average amount of time kids spend reading books to a known or assumed value.

c. The appropriate hypothesis test for this scenario would be a Paired t-Test, as we are comparing the performance of the same students on math problems with and without music.

d. The appropriate hypothesis test for this scenario would be a 2-Sample t-Test, as we are comparing the average age of professional baseball players to the average age of professional football players.

e. The appropriate hypothesis test for this scenario would be a 1-Prop z-Test, as we are testing the proportion of children with autism.

f. The appropriate hypothesis test for this scenario would be a 2-Prop z-Test, as we are comparing the proportions of immigrants in the US and Canada.

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Result of the problem   is  f = u + iv is a constant function on D.

To show that f is a constant function, we can use the Cauchy-Riemann equations. Since f is analytic on D, we know that it satisfies the Cauchy-Riemann equations, which state that u_x = v_y and u_y = -v_x.

Taking the partial derivative of u with respect to x and v with respect to y, we get:

u_xx = v_yx
and
v_yy = -u_xy

Since f is analytic, its second partial derivatives exist and are continuous. Therefore, we can substitute these equations into each other and get:

u_xx = -u_xy

Using the mixed partial derivative theorem, we know that u_xy = u_yx, so we can rewrite the above equation as:

u_xx = -u_yx

Since u and v are both real-valued functions, they are continuous on D. Therefore, we can apply the mean value theorem for partial derivatives to both sides of the above equation to get:

0 = u_xx(x,y) + u_yx(x,y) / 2

Since this holds for all (x,y) in D, we can conclude that u is a harmonic function on D. By Liouville's theorem, since u is a bounded harmonic function, it must be constant.

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The volume of a cylinder can be calculated using the formula:

V = πr^2h

where V is the volume, r is the radius, and h is the height.

First, we need to find the radius of the drum. The diameter is given as 14 cm, so the radius is half of that, or 7 cm.

Now we can plug in the values:

V = π(7 cm)^2(10 cm)

V = π(49 cm^2)(10 cm)

V = 1,539.38 cm^3 (rounded to two decimal places)

Therefore, the volume of the cylindrical drum of water is approximately 1,539.38 cubic centimeters.

The correct answer is:

c. The z-value for a standard normal distribution can be either positive or negative.

The z-value, also known as the standard score, measures the distance between a data point and the mean of its distribution in units of standard deviation. It is calculated by subtracting the population mean from the data point and then dividing the result by the standard deviation.

Since the mean of a standard normal distribution is zero, the z-value simply represents the number of standard deviations a data point is from the mean. As a result, the z-value can be either positive or negative, depending on whether the data point is above or below the mean, respectively.

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1. The area formula to use for each of the faces and base is the area of triangle

2. The surface area is 139.5 square yards

From the question, we have the following parameters that can be used in our computation:

The triangular pyramid

The above means that

The faces and the base of the figure are triangles

So, the area formula to use for each of the faces and base is the area of triangle formula

This is the sum of the areas of the shapes

So, we have

Surface area = 3 * 1/2 * 9 * 8 + 1/2 * 7 * 9

Evaluate

Surface area = 139.5

Hence, the surface area is 139.5 square yards

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The opportunity cost of 1 lunch box is 3 sandwich containers. This means workers are giving up the opportunity to produce 3 sandwich containers

The opportunity cost represents the value of the next best alternative forgone when making a choice. In this case, the workers at Shadowland have the option to produce either lunch boxes or sandwich containers.

Given that they can produce 6 lunch boxes or 18 sandwich containers per hour, we can calculate the opportunity cost.

To find the opportunity cost of 1 lunch box, we compare the number of sandwich containers that could have been produced in the same amount of time.

Since they can produce 18 sandwich containers per hour, the opportunity cost of 1 lunch box is the number of sandwich containers that could have been produced instead, which is 18/6 = 3.

Therefore, the opportunity cost of 1 lunch box is 3 sandwich containers. This means that for every lunch box produced, the workers are giving up the opportunity to produce 3 sandwich containers, which represents the trade-off in their production choices.

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The cost of tickets at the NC State Fair can be represented by a piecewise function that considers different age groups and their corresponding ticket prices.

Let's define a piecewise function, C(x), where x represents the age of the individual. The function will return the cost of the ticket for each age group. Here's the breakdown:

For adults aged 13-64, the ticket price is $10.

Therefore, for 13 ≤ x ≤ 64, C(x) = $10.

For children aged 6-12, the ticket price is $5.

Thus, for 6 ≤ x ≤ 12, C(x) = $5.

Children aged 5 and under can enter the fair for free.

Hence, for x ≤ 5, C(x) = $0.

Senior adults aged 65 and above also receive free admission.

Therefore, for x ≥ 65, C(x) = $0.

By using this piecewise function, you can easily determine the cost of tickets at the NC State Fair based on the age group of the individual attending.

For example, if someone is 25 years old, the cost of their ticket would be C(25) = $10.

Similarly, a 7-year-old child would have a ticket cost of C(7) = $5.

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The probability that x is less than 0.5 and y is greater than 0.6 is 0.0087.

The given joint probability density function of x and y is:

f(x,y) = {

x × y^8, 0 <= x <= 1, 0 <= y <= 1,

0, elsewhere

}

To determine the marginal probability density function of x, we integrate the joint probability density function over the y-axis:

f(x) = [tex]\int [0,1] x\times y^8 dy[/tex]

=[tex]x \times [y^{9/9}]_{[0,1]}[/tex]

= x/9

Similarly, to determine the marginal probability density function of y, we integrate the joint probability density function over the x-axis:

f(y) = [tex]\int[0,1] x \times y^8 dx[/tex]

= [tex]y^8 \times [x^{2/2}] _{[0,1]}[/tex]

= [tex]y^{8/2}[/tex]

To determine the probability that x is less than 0.5 and y is greater than 0.6, we use the joint probability density function and integrate over the given region:

P(x < 0.5 and y > 0.6) = [tex]\int[0.6,1] \int[0,0.5] x\times y^8 dx dy[/tex]

= [tex]\int[0.6,1] y^{8/2} \times [x^{2/2}][0,0.5] dy[/tex]

= [tex]\int[0.6,1] y^{8/16} dy[/tex]

= [tex][y^9/144][0.6,1][/tex]

= 0.0087

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The probability that x is less than 0.5 and y is greater than 0.6 is approximately 0.00011.

To determine the probability that x is less than 0.5 and y is greater than 0.6, we need to integrate the joint probability density function over the specified region.

Given the joint probability density function:

f(x, y) = {

x × y^8, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1,

0, elsewhere

}

To find the probability, we integrate the joint density function over the region:

P(x < 0.5 and y > 0.6) = ∫∫R f(x, y) dxdy

= ∫[0,0.5] ∫[0.6,1] (x × y^8) dy dx

= ∫[0,0.5] [((x × y^9)/9) |_0.6^1] dx

= ∫[0,0.5] (x/9 - (0.6^9 × x)/9) dx

= [(x^2)/18 - (0.6^9 × x^2)/18] |_0^0.5

= [(0.5^2)/18 - (0.6^9 × 0.5^2)/18] - [0 - 0]

= (1/72 - (0.6^9)/18) ≈ 0.00011

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Answer:

90$

Step-by-step explanation:

1. Find out what the question is asking

2. So now we have to find how much 18% of 500$ is

3. In conclusion, 18% commission of 500$ is 90$

The number of students in each row was 41.

In this case, since the square formation has the same number of rows and columns, we can represent both dimensions as 'x'. Therefore, the total number of students in the parade can be expressed as:

Total number of students = Number of rows × Number of columns

Given that there were 1681 students in the parade, we can substitute the values into the equation:

1681 = x × x

Now we have a quadratic equation. To solve for 'x', we can take the square root of both sides since the square root of a number times itself equals the number:

√1681 = √(x × x)

41 = x

Therefore, there were 41 students in each row of the square formation.

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The average throughput for this connection up through time = 5 RTT can be calculated using the formula: (N * MSS) / (5 * RTT).

To calculate the average throughput for this connection up through time = 5 RTT (round-trip time), you will need to follow these steps:

1. Determine the MSS (maximum segment size) and RTT for the connection. Since these values are not provided, I will use placeholders: MSS = X and RTT = Y.

2. Calculate the total time taken for the connection up through time = 5 RTT. In this case, the total time is 5 * Y, where Y is the RTT.

3. Determine the total amount of data transferred during this time. This would require information about the connection and the number of segments transmitted. Let's assume the connection transferred N segments during the 5 RTT period.

4. Calculate the total data transferred in terms of MSS. This is done by multiplying the number of segments (N) by the MSS (X): Total data = N * X.

5. Finally, calculate the average throughput by dividing the total data transferred by the total time taken: Average Throughput = (N * X) / (5 * Y).

In summary, the average throughput for this connection up through time = 5 RTT can be calculated using the formula: (N * MSS) / (5 * RTT).

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The total mass of the surface S with density given by xyz is (2√6/15).

To find the total mass of the surface S with density given by xyz, we need to evaluate the surface integral:

M = ∫∫S xyz dS

where dS is the surface area element.

We can parameterize the surface S using two variables u and v:

r(u, v) = (1 - u - v) (1, 0, 0) + u (0, 2, 0) + v (0, 1, 1)

where 0 ≤ u, v ≤ 1 and u + v ≤ 1.

The normal vector to the surface S at the point r(u, v) is given by the cross product of the partial derivatives of r with respect to u and v:

N(u, v) = ∂r/∂u × ∂r/∂v = (-2, 1, 2)

The magnitude of the normal vector is:

|N(u, v)| = √(2² + 1² + 2²) = √9 = 3

So the unit normal vector to the surface is:

n(u, v) = N(u, v) / |N(u, v)| = (-2/3, 1/3, 2/3)

The surface area element dS can be computed as the magnitude of the cross product of the partial derivatives of r with respect to u and v:

dS = |∂r/∂u × ∂r/∂v| du dv

= |(0, -2, 2) x (-1, 2, 1)| du dv

= |-4i - 2j - 4k| du dv

= 2√6 du dv

So the surface integral for the total mass becomes:

M = ∫∫S xyz dS = ∫0¹ ∫0(1-u) (x(u,v) y(u,v) z(u,v)) (2√6) dv du

where x(u,v) = 1 - u - v, y(u,v) = 2u, and z(u,v) = v.

Substituting these expressions into the integral, we get:

M = ∫0¹ ∫0(1-u) (1 - u - v)(2u)(v)(2√6) dv du

M = (4√6/3) ∫0¹ ∫0(1-u) (u - u² - uv)(v) dv du

M = (4√6/3) ∫0¹ [(u³/3) - (u⁴/4) - (u³/6) + (u⁴/4)] du

M = (4√6/3) ∫0¹ [(u⁴/4) - (u³/4)] du

M = (4√6/3) [(1/20) - (1/16)]

M = (2√6/15)

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Profit-maximizing refers to the level of output or production at which a business or a firm achieves the highest possible profit.

a) To find Sue's profit-maximizing level of output, we need to find the quantity where marginal revenue equals marginal cost. Marginal revenue is the derivative of the demand function, which is MR(y) = 100 - y/2. Marginal cost is the derivative of the cost function, which is MC(y) = 2y + 10. Setting MR(y) equal to MC(y) and solving for y, we get:

100 - y/2 = 2y + 10

90 = 5/2 y

y* = 36

So Sue's profit-maximizing level of output is 36.

b) To find the price at this level of output, we substitute y* into the demand function:

p* = 100 - (1/2)(36)

p* = $82

So the price at this level of output is $82.

c) To find Sue's profit, we need to subtract her total cost from her total revenue. Total revenue is price times quantity, or TR(y*) = p(y*) * y*:

TR(y*) = $82 * 36 = $2,952

Total cost is C(y*) = y*^2 + 10y*:

C(y*) = 36^2 + 10(36) = $1,296

So Sue's profit is:

(pi)* = TR(y*) - C(y*) = $2,952 - $1,296 = $1,656

So Sue's profit is $1,656.

d) Consumer surplus is the difference between the total value consumers place on a good and the amount they actually pay for it. At the profit-maximizing price and quantity, consumer surplus is:

CS = (1/2)(p* - MC(y*)) * y*

CS = (1/2)($82 - [2(36) + 10]) * 36

CS = $198

So the consumer surplus at the profit-maximizing price and quantity is $198.

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To determine the time it will take for the ball to return to the ground, we need to find the time when the ball reaches its maximum height and then double that time.

Given:

Initial velocity (u) = 120 feet per second

Acceleration due to gravity (g) = -32 feet per second squared (negative because it acts downward)

The equation of motion for the ball's height (h) as a function of time (t) can be expressed as:

h(t) = ut + (1/2)gt^2

When the ball reaches its maximum height, its vertical velocity (v) becomes 0. We can use this information to find the time it takes to reach the maximum height.

v = u + gt

0 = 120 - 32t

32t = 120

t = 120 / 32

t ≈ 3.75 seconds

The ball takes approximately 3.75 seconds to reach its maximum height. To find the total time of flight, we double this value:

Total time = 2 * 3.75

Total time ≈ 7.5 seconds

Therefore, it will take approximately 7.5 seconds for the ball to return to the ground.

To find out how many orange marbles are there in a bag containing 45 red marbles, given that each bag of marbles from Lashonda's Marble Company contains 8 orange marbles for every 5 red marbles, which is 72.

we can use the following steps:

Step 1: Determine the ratio of orange to red marbles in a bag from the given information. Each bag contains 8 orange marbles for every 5 red marbles. So the ratio of orange marbles to red marbles is 8:5. This means for every 8 orange marbles there are 5 red marbles. Therefore, the ratio of red marbles to orange marbles is 5:8

Step 2: Use the ratio of red to orange marbles to find how many orange marbles there are in a bag containing 45 red marbles. We can set up a proportion using the ratio of red marbles to orange marbles:5:8 = 45:xwhere x represents the number of orange marbles in the bag.Cross-multiplying, we get:5x = 8 × 45Simplifying:5x = 360Dividing both sides by 5:x = 72Therefore, a bag containing 45 red marbles has 72 orange marbles. Answer: 72.

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To find the distance from the building to the base of the ladder, we can use trigonometric functions.

Given:

The ladder length (hypotenuse) = 40 feet

The angle formed with the ground = 29.32°

We can use the sine function, which relates the length of the side opposite the angle to the hypotenuse:

sin(angle) = opposite / hypotenuse

In this case, the opposite side is the distance from the building to the base of the ladder.

sin(29.32°) = opposite / 40

To find the opposite side, we can rearrange the equation:

opposite = sin(29.32°) * 40

Using a calculator, we can evaluate the sine of 29.32°:

sin(29.32°) ≈ 0.4902

Now, we can calculate the distance from the building to the base of the ladder:

opposite ≈ 0.4902 * 40 ≈ 19.61 feet

Rounding to the nearest hundredth, the distance from the building to the base of the ladder is approximately 19.61 feet

Therefore, the correct answer is 19.59 feet.

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The simplified expression is [tex]-9z^32 + 3x^7y^4 / (z^5y^2).[/tex]

To simplify the expression [tex](36z^6^7 - 12x^7y^4) / (-4z^5y^2),[/tex] we can apply the rules of exponents and divide each term in the numerator by the denominator:

[tex](36z^6^7 - 12x^7y^4) / (-4z^5y^2)[/tex]

First, let's simplify the numerator: [tex]36z^6^7 - 12x^7y^4.[/tex]

Using the power of a power rule, we can simplify [tex]z^6^7 to z^(6*7) = z^42[/tex].

Therefore, the numerator becomes: [tex]36z^42 - 12x^7y^4.[/tex]

Now, we can divide each term in the numerator by the denominator:

[tex](36z^42 - 12x^7y^4) / (-4z^5y^2)[/tex]

= [tex]-36z^(42-5) / (4z^5) + 12x^7y^4 / (4z^5y^2)[/tex]

=[tex]-9z^37 / z^5 + 3x^7y^4 / (z^5y^2)[/tex]

Using the quotient rule of exponents, we subtract the exponents when dividing like bases:

= [tex]-9z^(37-5) + 3x^7y^4 / (z^5y^2)[/tex]

= -9z^32 + 3x^7y^4 / (z^5y^2)

Therefore, the simplified expression is [tex]-9z^32 + 3x^7y^4 / (z^5y^2).[/tex]

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Σ k/(2k+1) diverges. Σ (x^2+1)/k diverges.Σ (1000+k)/(3k+1) diverges.

Σ k ln(k) diverges. Σ k diverges. Σ ln(k) diverges. Σ k! diverges.

The divergence test states that if the limit of the nth term of a series is not zero as n approaches infinity, then the series must diverge. Using this test, we can determine whether the given series diverge or not.

For the first series, Σ k/(2k+1), as k approaches infinity, the limit of the nth term is 1/2, which is not zero. Therefore, the series diverges.

Similarly, for the second series, Σ (x^2+1)/k, the limit of the nth term is (x^2+1)/n, which does not approach zero as n approaches infinity. Therefore, the series diverges.

For the third series, Σ (1000+k)/(3k+1), as k approaches infinity, the limit of the nth term is 1/3, which is not zero. Therefore, the series diverges.

For the fourth series, Σ k ln(k), as k approaches infinity, the limit of the nth term is infinity, which is not zero. Therefore, the series diverges.

For the fifth series, Σ k, as k approaches infinity, the limit of the nth term is infinity, which is not zero. Therefore, the series diverges.

For the sixth series, Σ ln(k), as k approaches infinity, the limit of the nth term is infinity, which is not zero. Therefore, the series diverges.

For the seventh series, Σ k!, as k approaches infinity, the limit of the nth term is infinity, which is not zero. Therefore, the series diverges.

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