Find the critical points of f(x) = x - 18x² + 96x and use the Second Derivative Test (if possible) to determine whether each corresponds to a local minimum or maximum. (Use symbolic notation and fractions when needed)

the marginal profit when 4 units are produced and sold is approximately -$69.14. This means that when producing and selling the 4th unit, the profit will decrease by $69.14.

To find the marginal profit when 4 units are produced and sold, we first need to find the derivative of the profit function P(x) with respect to x. The given profit function is:

P(x) = (x^2) / (2x + 1)

Now, let's find its derivative, which represents the marginal profit function:

dP/dx = (d/dx(x^2))/(2x + 1) - (x^2)(d/dx(2x + 1))/((2x + 1)^2)

First, find the derivative of x^2 and 2x + 1:

d/dx(x^2) = 2x
d/dx(2x + 1) = 2

Now, substitute these values into the marginal profit function:

dP/dx = (2x)/(2x + 1) - (x^2)(2)/((2x + 1)^2)

Next, we'll find the marginal profit when 4 units are produced and sold. So, let's substitute x = 4 into the marginal profit function:

dP/dx(4) = (2 * 4)/(2 * 4 + 1) - (4^2)(2)/((2 * 4 + 1)^2)

dP/dx(4) = (8)/(9) - (32)(2)/(81)

dP/dx(4) = (8 - 64)/(81)

dP/dx(4) = -56/81

Since the profit is in hundreds of dollars, we need to multiply the marginal profit by 100 to get the value in dollars. Then, round to the nearest cent:

Marginal profit in dollars = (-56/81) * 100 ≈ -$69.14

So, the marginal profit when 4 units are produced and sold is approximately -$69.14. This means that when producing and selling the 4th unit, the profit will decrease by $69.14.

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The volume of the regular pentagonal prism with an edge length of 9 m and a height of 13 m is approximately 1811.6 m³ to the nearest tenth.

To find the volume of a regular pentagonal prism with an edge length of 9 m and a height of 13 m, follow these steps:

Step 1: Find the apothem (a) of the base pentagon. Use the formula a = s / (2 * tan(180/n)), where s is the edge length and n is the number of sides (5 for a pentagon).

a = 9 / (2 * tan(180/5))
a ≈ 6.1803 m

Step 2: Calculate the area (A) of the base pentagon. Use the formula A = (1/2) * n * s * a.

A = (1/2) * 5 * 9 * 6.1803
A ≈ 139.3541 m²

Step 3: Determine the volume (V) of the pentagonal prism. Use the formula V = A * h, where h is the height.

V = 139.3541 * 13
V ≈ 1811.6033 m³

So, the volume of the regular pentagonal prism with an edge length of 9 m and a height of 13 m is approximately 1811.6 m³ to the nearest tenth.

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we first need to find out how many street name boards can be purchased with a budget of R80,000. We can do this by dividing the budget by the cost of each street name board:

R80,000 ÷ R134 = 597.01

Since we cannot purchase a fraction of a street name board, we round down to the nearest whole number. Therefore, the municipality can purchase 597 new street name boards with a budget of R80,000.

To find out how much money will be left in the budget, we can subtract the cost of the street name boards from the initial budget:

R80,000 - (597 x R134) = R1,598

Therefore, the municipality will have R1,598 left in their budget after purchasing 597 new street name boards.

Street name boards are an important part of any municipality as they help residents navigate and locate specific areas. By budgeting for and installing new street name boards, the municipality is taking a proactive approach to improve their community's infrastructure and ensure the safety and well-being of their residents.

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Answer:

f(0) = 1

g(-2) = 3

f(-7)= und

g(4) x f(3) = -2 x 0 = 0

g(-4) = 2

g(x) = 0 --> x = 6, 0.5

f(x) = -1 --> x = -3, 5

f(g(3)) = f(-3) = -1

g(f(-2) = g(0) = -3

f(g(1)) = f(-3) = -1

f(g(5)) = f(-1) = 1

g(f(-4)) = g(-2) = 2

g(g(-6)) = g(4) = -2

g(f(0)) = g(1) = -3

g(f(-6)) = und

Step-by-step explanation:

In order to find the first group, such as f(0), you want to look at the f graph and find 0 on the x-axis. Wherever the y coordinate is will be the correct answer.

To find one such as f(g(3)), you want to dissect it like it is 2 problems. First, we want to find g(3) which is -3. Then we will find -3 on the f graph and find the answer with that y-coordinate.

The required value of te angle ∠3 = 100 degree.

A circle is a shape that is made up of all of the points in a plane that are at a certain distance from the center. Alternatively, it is the plane-moving curve traced by a point at a constant distance from a given point.

From the figure of the circle, we can identify that AD and CD are two diameter of the circle.

and ∠COA = ∠3 is inscribed angle made by up by 2 radian of the circle

So, arcCA = ∠3 = 100 degree

Thus, required value of te angle ∠3 = 100 degree.

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The mass of the thin bar with the given density function is 7/3.

To find the mass of the thin bar with the given density function p(x) = 2 + x^2 for 0 ≤ x ≤ 1:

You need to integrate the density function over the given interval.

Here are the steps to do that:

STEP 1: Set up the integral for mass:

Mass = ∫(density function) dx from x = 0 to x = 1
STEP 2:Plug in the given density function:

Mass = ∫(2 + x^2) dx from x = 0 to x = 1
STEP 3:Integrate the function with respect to x:

Mass = [2x + (x^3)/3] evaluated from x = 0 to x = 1
STEP 4: Evaluate the integral at the limits:

Mass = (2(1) + (1^3)/3) - (2(0) + (0^3)/3)
STEP 5: Simplify the expression:

Mass = (2 + 1/3) - 0
STEP 6: Calculate the final mass:

Mass = 2 + 1/3 = 7/3

So, the mass of the thin bar with the given density function is 7/3.\

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The answer is 96 inches, which is equivalent to 8 feet.

To find the length of the top edge of the yield sign, we need to use the formula for the area of a trapezoid:

A = (b1 + b2)h/2

where A is the area of the trapezoid, h is the height, b1, and b2 are the lengths of the two parallel bases of the trapezoid.

In this case, we are given the area of the sign (216 square inches) and the height (18 inches), but we don't know the length of either base. However, we do know that the shape of a yield sign is that of a regular octagon, which means it has eight equal sides and eight equal angles.

If we draw a line from the top of the sign to the midpoint of one of the sides, we will form a right triangle with the height of the sign as one leg, half the length of the top edge as the other leg, and the length of one of the sides as the hypotenuse. We can use the Pythagorean theorem to find the length of the side:

a^2 + b^2 = c^2

where a is the height of the sign (18 inches), b is half the length of the top edge (what we are trying to find), and c is the length of one of the sides.

Since the sign has eight sides, we can divide the total area by 8 to get the area of one of the eight triangles that make up the sign. We can then use this area to find the length of one of the sides:

A = (bh)/2

216 sq. in. = (bh)/2

432 sq. in. = bh

Since the sign is a regular octagon, each of the eight triangles has the same base (the side of the octagon) and height (half the length of the top edge), so we can use this equation to solve for b:

432 sq. in. = b(18 in.)/2

b = 48 in.

Now we know that half the length of the top edge is 48 inches, so the full length of the top edge is:

2(48 in.) = 96 in.

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Answer: 36

Step-by-step explanation:

[tex]\frac{1}{3} \pi 15^{2} h=2700\pi \\75h=2700\\h=36[/tex]

a) Approximately 79.38% of the phones have a battery life of at least 4 hours and 45 minutes.

b) Approximately 34.13% of the phones have a battery life between 4.5 hours and 5.25 hours.

c) Approximately 50% of the phones have a battery life less than 5 hours or greater than 5.5 hours.

a) What percentage battery life of 4 hours and 45 minutes?

a) For phones with a mean battery life of 5 hours and a standard deviation of 15 minutes, we can calculate the percentage of phones with a battery life of at least 4 hours and 45 minutes. By converting 4 hours and 45 minutes to minutes (4*60 + 45 = 285 minutes) and using the z-score formula, we find that the z-score is (285 - 300) / 15 = -1. Hence, the percentage is approximately 1 - 0.8359 = 0.1641, which is about 16.41%.

b) What percentage battery life between 4.5 hours and 5.25 hours?

b) To determine the percentage of phones with a battery life between 4.5 hours and 5.25 hours, we need to calculate the z-scores for both values. Converting the hours to minutes, we have 4.5 hours = 270 minutes and 5.25 hours = 315 minutes. The z-scores are (270 - 300) / 15 = -2 and (315 - 300) / 15 = 1. By referring to the standard normal distribution table, we find that the area between -2 and 1 is approximately 0.6141. Thus, the percentage is 0.6141 * 100 = 61.41%.

c) For the percentage of phones with a battery life less than 5 hours or greater than 5.5 hours, we need to calculate the z-score for both cases. The z-score for 5 hours is (300 - 300) / 15 = 0, and the z-score for 5.5 hours is (330 - 300) / 15 = 2. By referring to the standard normal distribution table, we find that the area to the left of 0 is 0.5 and the area to the right of 2 is 1 - 0.9772 = 0.0228. Adding these percentages, we get 0.5 + 0.0228 = 0.5228, which is approximately 52.28%.

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Answer:

Please look at the picture provided to see the correct triangle because each figure has no a,b, or c. Thanks

The length of the diagonal is 12. 7in

To determine the length of the diagonal, we need to know the Pythagorean theorem.

The Pythagorean theorem states that the square of the hypotenuse side is equal to the sum of the squares of the other two sides of a triangle.

The other two sides are the opposite and the adjacent sides.

From the information given in the diagram, we have that;

The opposite side = 12in

The adjacent side = 4in

Substitute the values

x² = 12² + 4²

find the squares

x² = 160

find the square root

x = 12. 7 in

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To find the coordinates of point B, we can use the section formula which states that the coordinates of the point that divides a segment with endpoints (x1, y1) and (x2, y2) in the ratio of m:n are given by:

((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n))

The coordinates of point B are (4.25, 8.25), and the answer is (D).

Here, A (2, 6) and C (5, 9) are the endpoints of the segment, and we want to partition the segment in the ratio of 3:1. So, we have:

m:n = 3:1

m+n = 4

Solving for m and n, we get:

m = 3, n = 1

Now, substituting  values in the section formula, we get:

((35 + 12)/(3+1), (39 + 16)/(3+1)) = (4.25, 8.25)

Therefore, the coordinates of point B are (4.25, 8.25), and the answer is (D).

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Answer:

(4.25, 8.25)

Step-by-step explanation:

i took the quiz

If f'(x) = 8x³ + 12x + 2 and f(1) = -4, f(-1) is equal to -18.

Given that f'(x) = 8x³ + 12x + 2, we can find the original function f(x) by integrating f'(x) with respect to x:

f(x) = 2x⁴ + 6x² + 2x + C, where C is an arbitrary constant.

We can then use the given initial condition f(1) = -4 to solve for C:

f(1) = 2(1)⁴ + 6(1)² + 2(1) + C = -4

Simplifying, we get:

C = -16

Therefore, the function f(x) is:

f(x) = 2x⁴ + 6x² + 2x - 16

To find f(-1), we substitute x = -1 into the expression for f(x):

f(-1) = 2(-1)⁴ + 6(-1)² + 2(-1) - 16 = -18

Thus, f(-1) equals -18.

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Answer: 1 cups of nuts : 1 1/2 cups of raisins

Step-by-step explanation:

Answer:

h = 24 cm

Step-by-step explanation:

Given:

C (base) = 60π cm

V (volume) = 21,600π cm^3

Find: h (height) - ?

[tex]c = 2\pi \times r[/tex]

[tex]2\pi \times r = 60\pi[/tex]

[tex]2r = 60[/tex]

[tex]r = 30[/tex]

We found the length of the radius

v = 1/3 × πr^2 × h

1/3 × π × 900 × h = 21600π

Multiply both sides by 3:

2700π × h = 64800π / : 2700π

h = 24 cm

The total cost of repayment will be $7,737.00, and Marvin will pay $937.00 in interest over the 5-year period.

To find the total cost of repayment, we need to calculate the total amount that Marvin will pay over the course of 5 years. Since he is making monthly payments, we need to first find the total number of payments he will make:

Total number of payments = 5 years x 12 months/year = 60 payments

The total amount Marvin will pay is then:

Total amount = 60 payments x $128.95/payment = $7,737.00

To find the total interest Marvin will pay, we need to subtract the original amount of the loan from the total amount he will pay:

Total interest = Total amount - Loan amount

Total interest = $7,737.00 - $6,800.00

Total interest = $937.00

Therefore, the total cost of repayment will be $7,737.00, and Marvin will pay $937.00 in interest over the 5-year period.

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The expressions that are equivalent to (3^3*3^-4)^-3 are (3^-1)^-3, 3^3 and 27

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

(3^3*3^-4)^-3

Evaluating the expression in the brackets using the law of indices, we have

(3^3*3^-4)^-3 = (3^-1)^-3

Next, we open the brackets

This gives

(3^3*3^-4)^-3 = 3^(-1 *-3)

When evaluated, we have

(3^3*3^-4)^-3 = 3^3

Evaluate the exponents

This gives

(3^3*3^-4)^-3 = 27

Hence, the expressions that are equivalent to (3^3*3^-4)^-3 are (3^-1)^-3, 3^3 and 27

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|4x +9|=|6-5x|

4x+5x=6-9

9x=-3

x=-3/9

x=-1/3

Answer:

x = 15

x = - ⅓

Step-by-step explanation:

|4x+9| = |6-5x|

4x+9=6-5x or 4x+9=-6+5x

Lets do the first one first

4x+9=6-5x

9x=-3

x= -3/9 = -1/3

4x+9=-6+5x

15=x

So the two solutions are x = 15 and x = -⅓

The singer sold approximately 512 music downloads for  17p and 456 CDs for 35p and earned a total profit of £246. 64.

Given data:

Total profit = £246. 64

The price for each music download is = 17p

Number of CDs sold = 456 CD

Price for each CD = 35p

Assume that the singer sold x music downloads. when she earns 17p for each music download then her total earnings from music downloads will be 0.17x. The total earnings from selling the CD are,

= 0.35 × 456

= 159.6.

From the above data, we can write the equation to find the value of x by,

0.17x + 159.6 = 246.64

0.17x = 246.64 - 159.6

0.17x = 87.04

x = 87.04/0.17

x = 512

Therefore, the singer sold approximately 512 music downloads.

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The median is a better center for this data because the data is not normally distributed and clusters on the left.

Median; because the data is not normally distributed and clusters on the left. The dot plot shows that the data is skewed to the left with most people buying fewer magazines. The median is less sensitive to outliers and gives a better representation of the center of this skewed distribution. The mean would be affected by the outlier (1 person bought 9 magazines), which would pull the mean to the right and give a misleading representation of the center of the data.

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The mean amount of D-glucose in cockroach hindguts is 44.24 micrograms, with a standard deviation of 26.16 micrograms. The 96% confidence interval for the mean is (20.19, 68.29) micrograms.

We will first calculate the mean, standard deviation, and then the 96% confidence interval for the given data on D-glucose amounts in cockroach hindguts.

Data: 55.95, 68.24, 52.73, 21.5, 23.78

1. The mean is calculated as follows.

Mean = (55.95 + 68.24 + 52.73 + 21.5 + 23.78) / 5 = 221.2 / 5 = 44.24

The mean amount of D-glucose in cockroach hindguts is approximately 44.24 micrograms.

2. To find the standard deviation, we calculate the variance first:

Variance = [(55.95 - 44.24)^2 + (68.24 - 44.24)^2 + (52.73 - 44.24)^2 + (21.5 - 44.24)^2 + (23.78 - 44.24)^2] / (5 - 1) = 2739.736 / 4 = 684.934

Standard Deviation = √684.934 = 26.16 (rounded to two decimal places)

The standard deviation of D-glucose in cockroach hindguts is approximately 26.16 micrograms.

3. To calculate the 96% confidence interval, we need to find the margin of error:

Margin of Error = (Critical value * Standard Deviation) / √sample size

For a 96% confidence interval, the critical value (z-score) is approximately 2.05.

Margin of Error = (2.05 * 26.16) / √5 ≈ 24.05

Now, we can find the confidence interval:

Lower limit: Mean - Margin of Error = 44.24 - 24.05 = 20.19

Upper limit: Mean + Margin of Error = 44.24 + 24.05 = 68.29

The 96% confidence interval for the mean amount of D-glucose in cockroach hindguts approximately is (20.19, 68.29) micrograms.

Note: The question is incomplete. The complete question probably is: To study the metabolism of insects, researchers fed cockroaches measured amounts of a sugar solution. After 2, 5, and 10 hours, they dissected some of the cockroaches and measured the amount of sugar in various tissues. Five roaches fed the sugar D-glucose and dissected after 10 hours had the following amounts (in micrograms) of D-glucose in their hindguts: Amounts of D-glucose 55. 95, 68. 24, 52. 73, 21. 5, 23. 78. The researchers gave a 96% confidence interval for the mean amount of D-glucose in cockroach hindguts under these conditions. The insects are a random sample from a uniform population grown in the laboratory. We therefore expect responses to be Normal. What is the mean, standard deviation, and confidence interval?

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Given that 27.5 is the bottom end of the box, the lower quartile's value equation (Q1) must fall between 27.5 and 30. Hence, the appropriate response is 37.

Since, A mathematical equation links two statements and utilizes the equals sign (=) to indicate equality.

In algebra, an equation is a mathematical assertion that proves the equality of two mathematical expressions.

For instance, in the equation 3x + 5 = 14, the equal sign separates the numbers by a gap. A mathematical formula may be used to determine how the two sentences on either side of a letter relate to one another. The logo and the particular piece of software are usually identical. like, for instance, 2x - 4 = 2.

Here, 25% of the data fall inside the lower quartile (Q1), which is represented by that number. Q1 is situated near the bottom of the box in the box plot.

According to the description, the box's boundary is at 30, and its size ranges from 27.5 to 42.5 on the number line. As a result, the median value of the middle 50% of the data is 37, with a range of 27.5 to 42.5.

There are some data points outside the middle 50% since the lines outside the box finish at 15 and 55.

Hence; 27.5 is the bottom end of the box, the lower quartile's value (Q1) must fall between 27.5 and 42.5.

Hence, the appropriate response is,

= 37

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Answer:

Step-by-step explanation:

False.

Solids with curves can be laid out as a net. In fact, there are many solids with curves that can be represented by a net. For example, a cylinder can be represented by a net consisting of two circles for the top and bottom faces and a rectangle wrapping around the circumference of the circles for the side face. Similarly, a cone can be represented by a net consisting of a circle for the base and a sector of a circle for the lateral face, which can be unfolded and attached to the circular base.

While it may be more difficult to visualize and create a net for a solid with curves compared to a solid with flat faces, it is still possible to do so in many cases.

There is a probability that 105 of the 350 students would select May 9 for the dance.

To solve the above we need to find the proportion of students in the sample who voted for each date and this can be done by:

May 2:  18/70

  = 0.257

May 9: 39/70

  = 0.557

May 16: 13/70

   = 0.186

if the whole population of 350 students voted, then

May 2: 0.257 x 350

   =  90

May 9: 0.557 x 350

 = 195

May 16: 0.186 x 350

  = 65

From the above calculation, we can see that if 105 students out of 350) were to choose a date, the most likely date that they will select is  May 9, since it is the one that has the highest proportion of votes in the sample.

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Answer:

324π

Step-by-step explanation:

Area of circle = r² · π

r = 18 in

Find its area in terms of pi.

We Take

18² · π = 324π

So, the area of the circle is 324π.

Answer:

A = 324π

Step-by-step explanation:

A = πr²

A = π(18)²

A = π(324)

A = 324π

The interest that has to be paid for the car is $ 4534.2.

Compound interest is a type of interest calculation in which the interest earned is added to the principal amount.

The principal is the sum that is left to be paid after the down payment hence;

Principal = $24, 599 - $5, 000

= $19599

A = P(1 + r/n)^nt

A = amount

P =principal

r = rate

n = Number of times compounded

t = time

Then we have that;

A = 19599( 1 + 0.0425)^5

A = $24133.2

Then ;

Interest = Amount - Principal

Interest = $24133.2 -  $19599

=$ 4534.2

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Answer:

The answer is -84,484

Step-by-step explanation:

using Bodmas

multiplication first

5774+252-(2586×35)

5774+252-90510

6026-90510

-84,484

In Amelie's situation, the remainder of 7 indicates that she has 7 muffins left over that cannot be packed into a full box of 9.

When Amelie divides the total number of muffins (385) by the number of muffins per box (9), she obtains a quotient of 42 and a remainder of 7. The quotient represents the number of complete boxes that Amelie can fill with 9 muffins each, while the remainder represents the number of muffins that cannot be put into a full box.

In other words, the quotient tells us how many full boxes Amelie can pack, and the remainder tells us how many muffins are left over after packing all the full boxes. In this case, Amelie can pack 42 full boxes, each with 9 muffins, which totals to 378 muffins. The remaining 7 muffins cannot fill a full box, and they are left over after all the full boxes are packed.

The "R 7" notation is commonly used to indicate the remainder in long division problems. The letter "R" stands for "remainder," and the number following it (in this case, 7) represents the actual remainder. The remainder is important because it indicates how many items are left over after dividing them into equal-sized groups.

In Amelie's situation, the remainder of 7 indicates that she has 7 muffins left over that cannot be packed into a full box of 9.

Overall, interpreting the "R 7" in this problem helps us understand how many full boxes of muffins Amelie can pack and how many muffins are left over after packing the full boxes.

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