What would cause a discontinuity on a rational function (a polynomial divided by another polynomial)?

Answer: x ≅ -0.9 or -1.8

Step-by-step explanation:

[tex]3(3x+4)^2 - 6 = 0[/tex]

[tex]3(3x+4)^2 = 6[/tex]

[tex]3(9x^2+24x+16) = 6[/tex]

[tex]9x^2+24x+16 = 2[/tex]

[tex]9x^2+24x+14 = 0[/tex]

Use the quadratric formula to get:

x ≅ -0.9 or -1.8

To determine this family's annual medical aid tax credit, we need to consider their medical aid expenses for the year. Medical aid expenses are expenses related to medical services that are not covered by the government or medical insurance.

This family can claim a tax credit of up to R310 per month for the main member and the first dependent, and R209 per month for each additional dependent. This tax credit is only applicable to registered medical schemes and is deducted from the tax payable.

In addition to medical aid expenses, this family will also need to consider the 15% VAT payable on all goods and services, except for sanitary pads, fresh produce, and a few other staple food items. This means that if this family spends R10,000 on goods and services, they will need to pay an additional R1,500 in VAT.

However, the good news is that they won't have to pay VAT on their fresh produce and staple food items. This will help to reduce their overall expenditure on food, which is an essential expense for every family.

In conclusion, while this family will need to pay VAT on most goods and services, they can claim a tax credit for their medical aid expenses. Additionally, they won't have to pay VAT on fresh produce and staple food items, which will help to reduce their overall food expenditure.

By carefully managing their expenses and taking advantage of tax credits and exemptions, this family can ensure that they are able to provide for their essential needs while also managing their financial obligations.

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By using  sum , difference , product , quotient and one that involves at least two operations that have the value of -3/4.

Now, We have to find the five expressions :

Expression for sum is -

1/4 + (-1) = -3/4

Expression for difference is -

(1/4 - 1) = -3/4

Expression for product is -

(-3/2)(1/2) = -3/4

Expression for quotient is -

(1 - 4)/4 = -3/4

Expression that involves at least two operations is -

-(1/4 + 2/4) = -3/4

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x = -5, dx/dt is equal to -2/25.

To find dx/dt, we need to use the chain rule of differentiation.

We know that dy/dt = -4 and we have the equation y = -5x^2 + 2.

Taking the derivative of both sides with respect to t, we get:

dy/dt = d/dt (-5x^2 + 2)

Using the chain rule, we can write this as:

dy/dt = (-10x) (dx/dt)

Now, we can plug in x = -5 and dy/dt = -4:

-4 = (-10(-5)) (dx/dt)

Simplifying, we get:

-4 = 50 (dx/dt)

Dividing both sides by 50, we get:

dx/dt = -4/50

Simplifying further, we get:

dx/dt = -2/25

Therefore, at x = -5, dx/dt is equal to -2/25.

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In this scenario, the domain would be the set of all possible values for the amount of time it takes Sammy Speedster to drive from Houston to San Antonio. Since he is driving on a daily basis, the domain could be considered a continuous range from zero to some maximum value, perhaps 10 hours.

It's possible that Sammy could complete the trip in less than 3 hours if he were driving at a very high speed, but this would not be a common occurrence. Therefore, a reasonable domain could be considered to be between 3 and 10 hours.

The range, on the other hand, would be the set of all possible distances that Sammy could drive in the allotted time. Since we know that the distance is a constant 200 miles, the range would simply be 200 miles. It's possible that Sammy could drive more than 200 miles in a day if he were assigned additional routes, but this would not be relevant to this scenario.

In summary, a reasonable domain for Sammy's logbook would be between 3 and 10 hours, and the range would be a constant 200 miles.

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To find the first term and common ratio of a geometric sequence, we can use the formula for the nth term:

a_n = a_1 * r^(n-1)

We are given the eighth and eleventh terms, so we can set up two equations:

a_8 = a_1 * r^(8-1) = 81920

a_11 = a_1 * r^(11-1) = 5242880

After dividing the second with by the first equation, we get:

(a_1 * r^(11-1)) / (a_1 * r^(8-1)) = 5242880 / 81920

Simplifying, we get:

r³ = 64

Doing the root of cube both sides, we get:

r = 4

Substituting this into the first equation, we get:

a_1 * 4^(8-1) = 81920

a_1 * 4^7 = 81920

a_1 = 5

Therefore, the first term is 5 and the common ratio is 4.

In a geometric sequence, each term is obtained by multiplying the previous term by a constant factor called the common ratio (r). The formula for the nth term of a geometric sequence is:

an = a * r^(n-1)

Given that the 8th term (a8) is 81,920 and the 11th term (a11) is 5,242,880, we can set up the following equations:

81920 = a * r^(8-1) => 81920 = a * r⁷ (1)

5242880 = a * r^(11-1) => 5242880 = a * r¹⁰ (2)

Now, we need to find the values of a (the first term) and r (the common ratio). Divide equation (2) by equation (1):

(5242880 / 81920) = (a * r¹⁰) / (a * r⁷)

64 = r^3

Now, we can find the common ratio r:

r = 4 (since 4³ = 64)

Next, substitute r back into equation (1) to find the first term a:

81920 = a * 4⁷

a = 81920 / 16384

a = 5

So, the first term (a) is 5, and the common ratio (r) is 4.

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This system represents the purchase made by Muriel (2 boxes of pencils and 4 erasers for $11) and Ramon (1 box of pencils and 3 erasers for $7).

The correct system of equations to find the price of one box of pencils (p) and one eraser (e) is:

2p + 4e = 11

p + 3e = 7

This system represents the purchase made by Muriel (2 boxes of pencils and 4 erasers for $11) and Ramon (1 box of pencils and 3 erasers for $7).

By solving this system of equations, you can determine the individual prices for a box of pencils and an eraser.

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If the parallelogram has an area of 25.2 cm² and the height is 4 cm, the length of the base is 6.3 cm.

To start, we know that the area of a parallelogram is given by the formula:

A = bh

where A is the area, b is the length of the base, and h is the height. We also know that the area of the parallelogram in this case is 25.2 cm² and the height is 4 cm.

Substituting these values into the formula, we get:

25.2 = b(4)

To solve for b, we can divide both sides by 4:

b = 25.2/4

b = 6.3

So the length of the base is 6.3 cm.

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Answer:

C

Step-by-step explanation:

All the other ones aren't increasing with the same proportion

Only C is increasing by same number each time (which is 26)

The perimeter of the base of the triangular prism =

The area of the base of the triangular prism =

The total surface area of the triangular prism =

A triangular prism is a three-dimensional geometric shape that consists of two parallel triangular bases connected by three rectangular or parallelogram faces. The faces that connect the two bases are called lateral faces. The lateral edges are the edges that connect the lateral faces, and the base edges are the edges that form the triangles.

The perimeter of the base of the triangular prism

p = 10cm + 10cm + 12cm = 32cm

The area of the base of the triangular prism =

base area = (1/2) bh

= 1/2 × b × h = 1/2 × 12 × 8 = 48cm².

The total surface area of the triangular prism =  (Perimeter of the base × Length of the prism) + (2 × Base Area)

= (32 + 34) + (2 × 48) = 66 + 96 = 162cm².

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Answer:

Modise is right.

This diagram does not represent a function because each input value does not correspond to exactly one output value. The input value 5 corresponds to two outputs, 2 and 9.

The probability that a student gets a lunch with chips and apple juice is 1/12, and the probability that a student gets a lunch without chips is 1/2.

There are 3 choices of sandwiches, 2 choices of sides, and 2 choices of drinks, so there are a total of 3x2x2 = 12 possible lunch combinations. To find the probability that a student gets a lunch that includes chips and apple juice, we need to count the number of lunch combinations that include chips and apple juice, and then divide by the total number of possible lunch combinations.

Number of lunch combinations that include chips and apple juice = 1 (chips and apple juice is only one combination)

Total number of possible lunch combinations = 12

Probability of getting a lunch that includes chips and apple juice = 1/12

To find the probability that a student gets a lunch that does not include chips, we need to count the number of lunch combinations that do not include chips, and then divide by the total number of possible lunch combinations,

Number of lunch combinations that do not include chips = 6 (3 choices of sandwiches x 2 choices of drinks) Total number of possible lunch combinations = 12, probability of getting a lunch that does not include chips = 6/12 = 1/2

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176, 264. your welcome!

The probability that four boys are given the first four seats on the first teacup is approximately equal to 0.004.

To determine the probability that four boys are given the first four seats on the first teacup.

The total number of ways to distribute 12 tickets among 12 seats is 12! (12 factorial), which is equal to 479,001,600.

We need to find the number of ways that four boys can be selected from the seven boys, multiplied by the number of ways that eight people (including the remaining three boys and five girls) can be selected from the ten remaining people,

multiplied by the number of ways that the selected people can be arranged on the teacup ride.

The number of ways to select four boys from seven boys is 7C4, which is equal to 35. The number of ways to select eight people from the remaining ten people is 10C8, which is equal to 45.

Finally, the number of ways to arrange the selected twelve people on the teacup ride is 4!, which is equal to 24.

Therefore, the probability that four boys are given the first four seats on the first teacup is (35 x 45 x 24) / 12!, which is approximately equal to 0.004.

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The given system of equations has one solution.

To determine the number of solutions of the functions. The given system over the interval (-3, 1], we need to find the intersection points of the two functions, f(x) and g(x), within that interval.

First, let's analyze each function separately:

The natural logarithm function ln(x) is only defined for positive                  values of x. In this case, we have ln(x + 3). To find the intersection points              with the interval (-3, 1], we need to ensure that x + 3 is positive.

For x in the interval (-3, 1], we have:

-3 < x ≤ 1

Adding 3 to both sides of the inequality:

0 < x + 3 ≤ 4

Therefore, the function f(x) = ln(x + 3) is defined over the interval (0, 4].

    2. Function g(x) = 2 * [tex]6^x[/tex]:

The exponential function [tex]6^x[/tex] is always positive for any real value of x. Multiplying it by 2 won't change the fact that the function remains positive. Hence, g(x) is positive for all real values of x.

Now, let's determine the intersection points of f(x) and g(x) within the interval (-3, 1].

Since g(x) is always positive and f(x) is defined over (0, 4], the intersection points occur where f(x) = g(x) > 0.

To solve this equation, we can rewrite it as ln(x + 3) - 2 * [tex]6^x[/tex] = 0.

Finding the exact solutions to this equation is not straightforward and may require numerical methods or graphing. However, it's clear that there is at least one solution within the interval (0, 4].

In conclusion, the given system has at least one solution over the interval (-3, 1].

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The main types of markets are:

So, the correct answer is "consumer" and "industrial".

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The percent of the phosphorus-32 decays each day is 0.56%.

The formula for the amount of radioactive isotope remaining after t days is given as:

y = a(0.5)^(t/14)

To find the percent of phosphorus-32 that decays each day, we need to find the fraction of the initial amount that decays each day. This can be found by subtracting the amount remaining after one day from the initial amount, and then dividing by the initial amount:

fraction decayed in one day = (a - a(0.5)^(1/14)) / a

Simplifying this expression gives:

fraction decayed in one day = 1 - (0.5)^(1/14)

To find the percent decayed in one day, we multiply by 100:

percent decayed in one day = 100(1 - (0.5)^(1/14))

Using a calculator, we get:

percent decayed in one day ≈ 0.56%

Therefore, the percent of phosphorus-32 that decays each day is approximately 0.56%.

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a. Sin A = opposite/hypotenuse = √15/4

b.  (Sin A + Cos B) ÷ 2 = (Sin A + Cos B)/2

c.  Tan A  = 1/5

The both angle could be of same size

Since Cos A = adjacent/hypotenuse,

Applying  the Pythagorean theorem,

we can find the opposite side:

opposite^2 + 1^2 = 4^2

opposite^2 = 16 - 1

opposite = √15

Now we can find the value of Sin A:

Sin A = opposite/hypotenuse = √15/4

b.

Sin A + Cos B = (Sin A) + (Cos B)

(Sin A + Cos B) ÷ 2 = (Sin A + Cos B)/2

c. Since Tan B = opposite/adjacent,  

we use  Pythagorean theorem to  find the hypotenuse:

hypotenuse^2 = 5^2 + 1^2

hypotenuse = √26

Tan A = opposite/adjacent = 1/5

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It takes about 1.5 seconds for the ball to hit the ground.

To find how long it takes for the ball to hit the ground, we need to find the value of x when h(x) = 0, since the height of the ball is 0 when it hits the ground. We can set -16x²+36 = 0 and solve for x:

-16x²+ 36 = 0

Dividing both sides by -16:

x² - 2.25 = 0

Adding 2.25 to both sides:

x²= 2.25

Taking the square root of both sides (we can ignore the negative root since time cannot be negative):

x = √(2.25) ≈ 1.5

Therefore, it takes about 1.5 seconds for the ball to hit the ground.

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A. The partial derivatives are not equal, F is not conservative, potential function f(x, y) = N

B. The partial derivatives are equal, F is conservative, potential function f(x, y) = -3xy - [tex]x^2[/tex] + C

C. The partial derivatives are not equal, F is not conservative, potential function f(x, y, z) = N

D. The partial derivatives are not equal, F is not conservative, potential function f(x, y, z) = N

E. The partial derivatives are not equal, F is not conservative, potential function f(x, y, z) = N

A. F(x, y) = (-6x – 7y) i +(-7x + 14y)j

To check if F is conservative, we compute the partial derivatives:

∂F/∂y = -6i - 7j

∂F/∂x = -7i + 14j

Since the partial derivatives are not equal, F is not conservative.

Potential function f(x, y) = N

B. F(x, y) = -3yi – 2xj

To check if F is conservative, we compute the partial derivatives:

∂F/∂y = -3i

∂F/∂x = -2j

Since the partial derivatives are equal, F is conservative.

Potential function f(x, y) =[tex]-3xy - x^2 + C[/tex], where C is a constant.

C. F(x, y, z) = -3xi – 2yj+k

To check if F is conservative, we compute the partial derivatives:

∂F/∂x = -3i

∂F/∂y = -2j

∂F/∂z = k

Since the partial derivatives are not equal, F is not conservative.

Potential function f(x, y, z) = N

D. F(x, y) = (-3 sin y)i + (-14y – 3x cosy)j

To check if F is conservative, we compute the partial derivatives:

∂F/∂y = -3cosy j - 14i

∂F/∂x = -3cosy j

Since the partial derivatives are not equal, F is not conservative.

Potential function f(x, y) = N

E. [tex]F(x, y, z) = -3xi - 7yj + 7z^2k[/tex]

To check if F is conservative, we compute the partial derivatives:

∂F/∂x = -3i

∂F/∂y = -7j

∂F/∂z = 14zk

Since the partial derivatives are not equal, F is not conservative.

Potential function f(x, y, z) = N

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The component of b onto a is (-1/3, 2/3, -1/3).

To find the component of b onto a, we first need to find the projection of b onto a. The projection of b onto a is given by the formula:

proj_a(b) = (b dot a / ||a||^2) * a

where dot represents the dot product and ||a|| represents the magnitude of vector a.

We can calculate the dot product of a and b as follows:

a dot b = (-2*1) + (4*0) + (2*3) = 4

We can calculate the magnitude of a as follows:

||a|| = sqrt((-2)^2 + 4^2 + 2^2) = sqrt(24) = 2sqrt(6)

Now we can plug these values into the formula for the projection of b onto a:

proj_a(b) = (b dot a / ||a||^2) * a
proj_a(b) = (4 / (2sqrt(6))^2) * (-2, 4, 2)
proj_a(b) = (4 / 24) * (-2, 4, 2)
proj_a(b) = (-1/3, 2/3, -1/3)

Finally, the component of b onto a is simply the projection of b onto a:
comp_a(b) = (-1/3, 2/3, -1/3)
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The temperature are the times  indicated are:

T( 0) =   5.800°

T( 0.22) = -4.9094 °

T (0.44) = -2.6792 °

T(0.66) = 2.3244°

T(0.88)  = 4.8184°

T(1.1) = -4.1686 °

To solve the above, we need to use the formuala that we given which is T(x) = 5.8cos(3.8πx)

Entering the various values of x, can can obtain

T(0) = 5.8 cos (3.8π  (0) )  = 5.8cos(0) = 5.800

T(0.22) = 5.8 cos (3.8  π(0.22)) ≈ -4.9094

T(0.44) = 5.8cos (3.8π (0.44) ) ≈ -  2.6792

T(0.66) = 5.8cos(3.8π (0.66)) ≈ 2.3244

T(0.88) = 5.8cos(3.8π(0.88)) ≈ 4.8184

T(1.1) = 5.8 cos(3.8π(1.1  )) ≈ -4.1686

So the above answers are correct.

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The equation that represents the Ferrells' savings after x months is y = 150x

In this equation, x represents the number of months that the Ferrells have been saving, and y represents the amount of money they have saved after x months.

The coefficient 150 represents the amount of money the Ferrells save each month, and it is multiplied by the number of months x to get the total savings y. For example, after 5 months of saving, the Ferrells would have saved:

y = 150(5) = $750

This equation can be used to calculate their savings at any point in time, as long as they continue to save $150 each month.

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The volume of a trapezoidal prism is 2362.5.

The total surface area​ is 852.

We have,

The volume of a trapezoidal prism.

V = ((a + b) / 2) × h × l

where:

a and b are the lengths of the two parallel sides (the bases) of the trapezoid

h is the height of the trapezoid (the perpendicular distance between the two bases)

l is the length of the prism (the distance between the two trapezoidal faces)

Now,

a = 9

b = 12

l = 15

Height h can be calculated using the Pythagorean theorem.

15² = (12 - 9)² + h²

h² = 225 + 9

h² = 234

h = √234

h = 15

Now,

The volume of a trapezoidal prism.

V = ((a + b) / 2) × h × l
V = ((9 + 12) / 2) x 15 x 15

V = 2362.5

And,

The surface area (A) of a trapezoidal prism can be calculated using the formula:

A = ph + 2B

where p is the perimeter of the trapezoidal base, h is the height of the prism, and B is the area of one of the bases.

So,

p = 12 + 8 + 12 + 8 = 44

h = 15

B = 12 x 8 = 96

Now,

Total surface area​.
= 44 x 15 + 2 x 96

= 660 + 192

= 852

Thus,

The volume of a trapezoidal prism is 2362.5.

The total surface area​ is 852.

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The length and width of the chocolate bar is 14.5 feet and 7.6 feet.

What is length?

Length is a measure of the size of an object or distance between two points. It refers to the extent of something along its longest dimension, or the distance between two endpoints.

What is width?

Width refers to the measure of the distance from one side of an object to the other side, perpendicular to the length. In geometry, width is usually measured in units such as meters, feet, or inches.

According to the given information:

Let's start by assigning variables to represent the length and width of the chocolate bar. Let x be the width of the chocolate bar in feet. Then, according to the problem:

The length of the chocolate bar is 0.3 feet shorter than twice its width, which means the length is (2x - 0.3) feet.

The base area of the chocolate bar is given as 61.04 square feet. We can use this information to set up an equation:

x(2x - 0.3) = 61.04

Expanding the left side of the equation:

2x^2 - 0.3x = 61.04

Moving all the terms to one side of the equation:

2x^2 - 0.3x - 61.04 = 0

Now we can solve for x using the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

where a = 2, b = -0.3, and c = -61.04. Substituting these values and simplifying:

x = (0.3 ± sqrt(0.3^2 + 4(2)(61.04))) / (2(2))

x ≈ 7.6 or x ≈ -4.0

Since the width of the chocolate bar cannot be negative, we can discard the negative solution. Therefore, the width of the chocolate bar is approximately 7.6 feet.

To find the length, we can use the equation we set up earlier:

length = 2x - 0.3

Substituting x = 7.6:

length = 2(7.6) - 0.3

length ≈ 14.5

Therefore, the length of the chocolate bar is approximately 14.5 feet and the width is approximately 7.6 feet.

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The solution set for the equation 1/2 = 1/3 - |(x-3/6)+x| is {1/12, -1/12}.

To solve the equation 1/2 = 1/3 - |(x-3/6)+x|, we need to isolate the absolute value expression and solve for x in two cases.

Case 1: (x-3/6)+x is nonnegative, which means the absolute value can be removed.

1/2 = 1/3 - (x-3/6)-x

1/2 = 1/3 - 2x + 1/2

2x = 1/6

x = 1/12

Case 2: (x-3/6)+x is negative, which means the absolute value must be flipped.

1/2 = 1/3 + (x-3/6)+x

1/2 = 1/3 + 2x - 1/2

2x = -1/6

x = -1/12

Therefore, the solution set is {1/12, -1/12}.

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The function f(x) = –2x^3 + 39x^2 -216x + 6 has one local minimum at x = 4 and one local maximum at x = 9.

To determine if the function f(x) = –2x^3 + 39x^2 -216x + 6 has a local minimum or maximum, we need to find the critical points of the function and then determine the nature of those critical points.

First, we take the derivative of the function to find the critical points:

f(x) = –2x^3 + 39x^2 -216x + 6

f'(x) = –6x^2 + 78x - 216

f'(x) = –6(x^2 - 13x + 36)

f'(x) = –6(x - 4)(x - 9)

Setting f'(x) = 0, we get:

–6(x - 4)(x - 9) = 0

This gives us two critical points at x = 4 and x = 9.

To determine the nature of these critical points, we need to look at the sign of the derivative on either side of each critical point.

When x < 4, we have:

f'(x) = –6(x^2 - 13x + 36) < 0

When 4 < x < 9, we have:

f'(x) = –6(x^2 - 13x + 36) > 0

When x > 9, we have:

f'(x) = –6(x^2 - 13x + 36) < 0

This means that f(x) is decreasing on the interval (–∞, 4), increasing on the interval (4, 9), and decreasing on the interval (9, ∞). Therefore, we have a local minimum at x = 4 and a local maximum at x = 9.

To confirm this, we can evaluate the function at these critical points:

f(4) = –2(4)^3 + 39(4)^2 -216(4) + 6 = –26

f(9) = –2(9)^3 + 39(9)^2 -216(9) + 6 = 603

Therefore, the function f(x) = –2x^3 + 39x^2 -216x + 6 has one local minimum at x = 4 and one local maximum at x = 9.

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In order to evaluate h'(9), we need to use the product rule, which states that the derivative of a product of two functions is equal to the first function times the derivative of the second function, plus the second function times the derivative of the first function. Mathematically, this can be expressed as:

(h(x))' = f(x)g'(x) + g(x)f'(x)

Using the given values, we can substitute them into the formula and solve for h'(9):

h'(x) = f(x)g'(x) + g(x)f'(x)
h'(9) = f(9)g'(9) + g(9)f'(9)
h'(9) = 9(2) + 3(-1.5)
h'(9) = 18 - 4.5
h'(9) = 13.5

Therefore, the value of h'(9) is 13.5.

In simpler terms, the product rule tells us that when we have a function that is the product of two other functions, we can find the derivative of that function by multiplying one function by the derivative of the other and adding it to the other function multiplied by the derivative of the first. In this case, we have two functions f(x) and g(x), and we use their respective values and derivatives to find the derivative of their product h(x).

MORE QUESTION ON product rule DERIVATIVE : https://brainly.com/question/30340084

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The slope of the line represent Parking cost $30 per day. The correct answer is C.

The slope of a line represents the rate of change between two variables. In this case, the line represents the relationship between Hector's parking fees and the number of days he parked his SUV.

The unit of the slope of the ine represents

Cost/ number of days = 30 $/day

The fact that the slope is negative (-$30) means that for each additional day Hector parked his SUV, his parking fees decreased by $30. This indicates that the parking fee is a function of the number of days parked, and it costs $30 per day. The correct option is C.

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--The given question is incomplete, the complete question is given

" Hector parked his SUV in long-term parking while he traveled to China. What does the slope of the line represent?

A Hector's parking fees decreased by $30 each week.

B Parking cost a flat fee of $30.

C Parking cost $30 per day.

D Hector got a $30 discount to park his SUV. ​ "--