6. Aisha's test scores are 74, 86, 67, and 76.
What score does she need on the last test in
order to average 80 on her tests?

Answer:

x^3 - 6x^2 + 12x - 8 = 0

Step-by-step explanation:

Let the roots of the equation x^3 - 8x + 3 = 0 be a, b, and c. Then, we know that the squares of the roots are a^2, b^2, and c^2.

By Vieta's formulas, we know that the sum of the roots of x^3 - 8x + 3 = 0 is 0, since there is no x^2 term:

a + b + c = 0

We also know that the product of the roots is 3/1 = 3:

abc = 3

Using the fact that the squares of the roots are a^2, b^2, and c^2, we can write:

a^2 + b^2 + c^2 = (a + b + c)^2 - 2(ab + ac + bc)

Since a + b + c = 0, this simplifies to:

a^2 + b^2 + c^2 = -2(ab + ac + bc)

Now, let's try to find a cubic equation with roots a^2, b^2, and c^2. We can start with:

(x - a^2)(x - b^2)(x - c^2) = 0

Expanding the left side, we get:

x^3 - (a^2 + b^2 + c^2)x^2 + (a^2b^2 + b^2c^2 + c^2a^2)x - a^2b^2c^2 = 0

Using the identity we found earlier for a^2 + b^2 + c^2, we can simplify this to:

x^3 + 2(ab + ac + bc)x - 3a^2b^2c^2 = 0

Now, we can substitute in the values we know for ab, ac, and bc:

ab = (a + b + c)(ab + ac + bc) - (a^2b + ab^2 + c^2a)

ac = (a + b + c)(ab + ac + bc) - (a^2c + ac^2 + b^2a)

bc = (a + b + c)(ab + ac + bc) - (b^2c + bc^2 + a^2b)

Plugging in these values and simplifying, we get:

x^3 - 6x^2 + 12x - 8 = 0

Therefore, the cubic equation we're looking for is:

x^3 - 6x^2 + 12x - 8 = 0

And the roots of this equation are the squares of the roots of x^3 - 8x + 3 = 0.

Answer:3

Step-by-step explanation:

Answer:

The equation of the line is in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.

From the given equation, we can see that the y-intercept is 19. This means that the line passes through the point (0, 19).

Therefore, the coordinates of the y-intercept are (0, 19).

The y-intercept of the equation y=6x+19 is the value of y when x equals 0, which is at the point (0,19).

Therefore, in the equation y=6x+19, the y-intercept is at the point (0, 19). Hence, the coordinates of the y-intercept are (0,19).

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There were 8 children in the classroom, and the initial guess of 7 children was incorrect.

Let's assume there were 7 children in the classroom. Since each child has one head and two legs, the number of heads would still be 7, and the number of legs would be 7 x 2 = 14.

Now we need to find out how many dogs there are. The total number of heads in the classroom is 14, so the number of dogs is 14 - 7 = 7. Each dog has one head and four legs, so the number of legs from the dogs is 7 x 4 = 28.

The total number of legs in the classroom is 40, so the total number of legs from the children and dogs is 14 + 28 = 42. Since there are only 40 legs, our initial assumption was incorrect.

To find the correct number of children in the classroom, we need to use algebra. Let c be the number of children, and let d be the number of dogs. We know that c + d = 14 and 2c + 4d = 40. We can solve for c by substituting the first equation into the second equation and simplifying:

2c + 4(14 - c) = 40

2c + 56 - 4c = 40

-2c = -16

c = 8

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The entire area is under the standard normal distribution curve, which is 1 by definition.

In other words, a typical normal random variable has a 100% chance of landing anywhere within its range of potential values (-∞ to +∞).

This makes sense since the area under the standard normal distribution curve, which is a continuous probability distribution, shows the likelihood that the random variable will take on any value within its range. Since the entire area under the curve is 1, the likelihood that any potential value for the random variable will occur is 1, or 100%.

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The probability of getting exactly 9 of them major in STEM is 0.139 or 13.9%.

The probability of getting at most 11 of them major in STEM is approximately 0.898 or 89.8%.

The probability of getting at least 11 of them major in STEM is approximately 0.318 or 31.8%.

a. Exactly 9 of them major in STEM.

In this case, the probability of success (a student majoring in STEM) is 0.27, and the number of trials is 35. The probability of exactly 9 students majoring in STEM is then given by the formula:

P(X = 9) = (35 choose 9) x (0.27)⁹ x (0.73)²⁶

where (35 choose 9) is the number of ways to choose 9 students out of 35, and (0.27)⁹ x (0.73)²⁶ is the probability of 9 successes and 26 failures. Evaluating this expression gives a probability of approximately 0.139 or 13.9%.

b. At most 11 of them major in STEM.

To calculate the probability that at most 11 students out of 35 major in STEM, we can use the cumulative binomial probability distribution. This distribution calculates the probability of at most X successes, where X is any number from 0 to the total number of trials.

The probability of at most 11 students majoring in STEM can be calculated as follows:

P(X <= 11) = P(X = 0) + P(X = 1) + ... + P(X = 11) = 0.898 or 89.8%.

c. At least 11 of them major in STEM.

The probability of less than 11 students majoring in STEM can be calculated using the cumulative binomial probability distribution, as in part (b). Specifically:

P(X < 11) = P(X = 0) + P(X = 1) + ... + P(X = 10)

Subtracting this probability from 1 gives the probability of at least 11 students majoring in STEM:

P(X >= 11) = 1 - P(X < 11) = 31.8%

Again, we can use the binomial distribution formula from part (a), or a binomial probability calculator or statistical software package to calculate this probability.

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Answer: 1:4

Step-by-step explanation:

In order to do this, we have to know that the octagon has an area which is four times less than the other area. Because of this, it shuld be 1:4

Answer:

a = 48

b = 28

Step-by-step explanation:

b = 0.5 × 56

b = 28

52 = 0.5 × (56 + a)

104 = 56 + a

a = 48

cheap answer, put them all in slope-intercept form, once we can see their slope, we know what's cooking.

same slope? likely parallel
different slope? likely neither

21)

we can rewrite them as

y = x + 6 and x +2 = y

slope in each is the same, 1, but the y-intercept differs, one is 6 the other 2, so they're parallel and a few units from each other.

22)

we can rewrite that as

[tex]\cfrac{x-8}{2}=y\implies \cfrac{1}{2}x-4=y\hspace{7em}y=-2x+1[/tex]

so one has a slope of 1/2 and the other -2 hmmm, let's check for the negative reciprocal of 1/2

[tex]\stackrel{~\hspace{5em}\textit{perpendicular lines have \underline{negative reciprocal} slopes}~\hspace{5em}} {\stackrel{slope}{ \cfrac{1}{2}} ~\hfill \stackrel{reciprocal}{\cfrac{2}{1}} ~\hfill \stackrel{negative~reciprocal}{-\cfrac{2}{1} \implies -2}}[/tex]

ahaa!!, so one slope is the negative reciprocal of the other, they are perpendicular.

23)

[tex]\cfrac{4x-9}{3}=y\implies \cfrac{4}{3}x-3=y\hspace{7em}\cfrac{3x-36}{4}=y\implies \cfrac{3}{4}x-9=y[/tex]

so one has a slope of 4/3 and the other of 3/4, neither.

24)

two horizontal lines, one above the other, parallel.

Answer:

parallel, perpendicular, neither , parallel

Step-by-step explanation:

• Parallel lines have equal slopes

• Product of slopes of perpendicular lines = - 1

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

21

y = x + 6 ← is in slope- intercept form , with slope m = 1

x - y = 2 ( subtract x from both sides )

- y = - x + 2 ( multiply through by - 1 )

y = x - 2 ← in slope- intercept form with slope = 1

Since the slopes are equal the lines are parallel

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

22

x - 2y = 8 ( subtract x from both sides )

- 2y = - x + 8 ( divide through by - 2 )

y = [tex]\frac{1}{2}[/tex] x - 4 ← in slope- intercept form , with slope m = [tex]\frac{1}{2}[/tex]

y = - 2x + 1 ← in slope- intercept form, with slope m = - 2

then [tex]\frac{1}{2}[/tex] × - 2 = - 1

since the product is - 1 the lines are perpendicular

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

23

4x + 3y = 9 ( subtract 4x from both sides )

3y = - 4x + 9 ( divide through by 3 )

y = - [tex]\frac{4}{3}[/tex] x + 3 ← in slope- intercept form , with slope = - [tex]\frac{4}{3}[/tex]

3x + 4y = 36 ( subtract 3x from both sides )

4y = - 3x + 36 ( divide through by 4 )

y = - [tex]\frac{3}{4}[/tex] x + 9 ← in slope- intercept form, with slope m = - [tex]\frac{3}{4}[/tex]

since slopes are neither equal nor product = - 1

The the lines are neither parallel nor perpendicular

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

24

the equation of a horizontal line, parallel to the x- axis is

y = c ( c is the value of the y- coordinates the line passes through )

y = 5 and y = - 2 are both in this form , thus are parallel lines

The slope of f(x) is smaller than the slope of g(x).

For a linear function that passes through two points (x₁, y₁) and (x₂, y₂) the slope is given by the formula:

a = (y₂ - y₁)/(x₂ - x₁)

Then the slope of g(x) is:

a = (3 - 0)/(0 + 2) = 3/2

And for f(x) the slope is 4/5, then we can see that:

3/2 > 4/5

Thus the slope of g(x) is larger.

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The volumes of the following shapes include:

728.00 m³.

135.78 yd³.

1260.00 cm³.

452.39 in³.

1344.00 m

1. Pyramid:

The formula for the volume of a pyramid is:

V = (1/3) × B × h

where B is the area of the base and h is the height.

For a pyramid with a rectangular base, the area of the base is:

B = l × w

where l and w are the length and width of the base, respectively.

Using the given measurements:

l = 13 m

w = 8 m

h = 21 m

Therefore, the area of the base is:

B = l × w = 13 m × 8 m = 104 m²

And the volume of the pyramid is:

V = (1/3) × B × h = (1/3) × 104 m² × 21 m ≈ 728 m³

Rounded to the nearest hundredth, the volume of the pyramid is 728.00 m³.

2. Cylinder:

The formula for the volume of a cylinder is:

V = πr²h

where r is the radius of the base and h is the height.

Using the given measurements, we have:

r = 3.5 yd / 2 = 1.75 yd

h = 10 yd

Therefore, the volume of the cylinder is:

V = πr²h = π × (1.75 yd)² × 10 yd ≈ 135.78 yd³

Rounded to the nearest hundredth, the volume of the cylinder is 135.78 yd³.

3. Cuboid:

The formula for the volume of a cuboid is:

V = l × w × h

Using the given measurements, we have:

l = 14 cm

w = 18 cm

h = 5 cm

Therefore, the volume of the cuboid is:

V = l × w × h = 14 cm × 18 cm × 5 cm = 1260 cm³

Rounded to the nearest hundredth, the volume of the cuboid is 1260.00 cm³.

4. Cone:

The formula for the volume of a cone is:

V = (1/3) × πr²h

where r is the radius of the base and h is the height.

Using the given measurements:

r = 12 in / 2 = 6 in

h = 37 in

Therefore, the volume of the cone is:

V = (1/3) × πr²h = (1/3) × π × (6 in)² × 37 in ≈ 452.39 in³

Rounded to the nearest hundredth, the volume of the cone is 452.39 in³.

5. Prism:

The formula for the volume of a prism is:

V = Bh

where B is the area of the base and h is the height.

Using the given measurements:

B = 6 m × 16 m = 96 m²

h = 14 m

Therefore, the volume of the prism is:

V = Bh = 96 m² × 14 m = 1344 m³

Rounded to the nearest hundredth, the volume of the prism is `1344.00 m

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The selling price of the antique through an online auction website is $680.

Given that,

Kyle sold an antique through an online auction website.

The website host charges Kyle $15, plus 2.5% of the final selling price of the antique.

After selling the antique, Kyle had to pay the website host $32.

Let x be the selling price of the antique.

We get an equation,

15 + (2.5% × x) = 32

15 + 0.025x = 32

0.025x = 17

x = $680

Hence the selling price is $680.

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The recursive relation of the data distribution a(n) = a ( n - 1 ) x n - ( n - 1 ).

A sequence of values is defined by a mathematical equation termed as a recursive relation, also known as recursive relation. It derives current value(s) through specified initial value(s) and previous value dependent rule(s). In essence, it generates numeric sequences.

Note that every individual unit can be derived by multiplying the preceding one with a rising integer, and subsequently subtracting a descending integer:

1 x 1 - 0 = 1

1 x 2 - 1 = 1

1 x 3 - 2 = 5

5 x 4 - 3 = 17

17 x  5 - 4 = 71

71 x 6 - 5 = 247

We can deduce the relation of a(n) = a ( n - 1 ) x n - ( n - 1 ).

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Answer:

Step-by-step explanation:

Using the given calculator, the product of (1.466 × 10–8) and (5.02 × 105) was found to be 7.3593 × 10–3. The coefficient of the product is 7.3593, which is obtained by multiplying the coefficients of the two given numbers. The exponent of the power in the product is –3, which is obtained by adding the exponent of the first number to the exponent of the second number and then subtracting it from the power of 10. Therefore, the final answer is 7.3593 × 10–3.

Answer:

Step-by-step explanation:

Using a calculator to find the product of two or more numbers is a simple process. First, enter the first number into the calculator using the number keys. Then, press the multiplication symbol (usually represented by the "x" key) followed by the second number. If there are more numbers to multiply together, repeat the process until all numbers have been entered. Finally, press the equals sign (often represented by the "=" key) to display the product of all the numbers. Remember to check the order of operations if necessary, and to double-check your input and output to avoid errors.

To multiply two numbers in scientific notation, we first multiply their coefficients (1.466 x 5.02 = 7.358). Then, we add their exponents together (-8 + 5 = -3). Therefore, (1.466 x 10⁻⁸) multiplied by (5.02 x 10⁵) equals 7.358 x 10⁻³ or 0.007358. The final answer has the exponent of -3, indicating that the answer should be moved 3 decimal places to the left to convert it from scientific notation to standard notation.

The simplified expression is -7.1y - 7.8x + 14.8.

The given expression is 2.6y + 14.8 - 7.8x - 9.7y. To combine like terms, we first identify the terms that have the same variable with the same exponent. In this case, we have two terms with y, and they both have the exponent 1. So, we can combine these terms together by adding their coefficients, which are 2.6 and -9.7.

2.6y - 9.7y can be simplified as (2.6 - 9.7)y, which is equal to -7.1y. Therefore, the expression can be written as:

-7.1y - 7.8x + 14.8

We cannot combine any other terms in this expression since they do not have the same variable or exponent.

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The surface area of the prism is 112 in²

A prism is a solid shape that is bound on all its sides by plane faces.

The surface area of a prism is expressed as;

SA = 2B +pH

where h is the height and B is the base area and p is the perimeter of the base

The perimeter of the base = 2(l+b)

= 2(3+4) = 2×7 =14 in

The area of the base = l× b

= 4× 3 = 12 in²

height = 7

Therefore the surface area = 2 × 12 + 14 × 7

= 14+98

= 112 in²

therefore the surface area of the prism is 112 in²

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Answer:

The amount of money the painter earns, in dollars, depends on the number of rooms he paints.

For each room painted, the painter charges $25. Therefore, the amount of money earned, y, is given by the equation:

y = 25x + 100

where x is the number of rooms painted and 100 is the cost of supplies.

To graph this equation, we can plot a point on the y-axis where x=0 (the fixed cost of supplies) at (0, 100), and then use the slope of 25 to find additional points. For example, when x=1, y=125; when x=2, y=150; and so on.

With this information, we can eliminate options (b) and (d) because they both show negative slopes, which would mean the painter loses money as he paints more rooms.

Option (c) shows a positive slope, but the line goes through the point (0,100) and (4,200), which means the painter earns $200 after painting 4 rooms, but this contradicts the given information that the painter charges $25 per room. Therefore, option (c) is also incorrect.

Option (a) correctly shows the slope of 25 and goes through the point (0, 100), and the line reaches the x-axis when x=4, indicating that the painter earns $0 after painting 4 rooms. This matches the given information that the painter charges $25 per room. Therefore, the correct answer is option (a).

Therefore, the graph that shows the relationship between the amount of money the painter earns, in dollars, and the number of rooms he paints is:

a coordinate grid with the x axis labeled rooms painted and the y axis labeled amount of money earned and a line going from the point 0 comma 100 through the point 4 comma 0

Answer:

The correct answer is a C. coordinate grid with the x axis labeled rooms painted and the y axis labeled amount of money earned and a line going from the point 0 comma 100 through the point 4 comma 200.

The painter pays $100 for supplies, so he will only earn money for the rooms he paints after he has recouped that cost. The first room he paints will earn him $25, the second will earn him $50, and so on. After he has painted 4 rooms, he will have earned $200.

The graph of this relationship would be a line that starts at the point (0, 100) and goes up to the point (4, 200).

Step-by-step explanation:

Answer: herefore, the average cost of the items Roy bought is $14.07.

Step-by-step explanation:

To find the average cost of the items Roy bought, we need to add up the total cost of all his purchases and divide by the total number of purchases.

Strategy:

1. Add up the cost of all Roy's purchases.

2. Count the number of purchases.

3. Divide the total cost by the number of purchases to find the average cost.

Explanation:

We will use this strategy because it is a straightforward way to find the average cost of multiple items. We will add up all the costs and divide by the number of purchases to get the average cost.

Calculations:

Total cost = (4 x $3.55) + $10.76 + $15.35 + $55.27 + $16.98

Total cost = $14.20 + $10.76 + $15.35 + $55.27 + $16.98

Total cost = $112.56

Number of purchases = 4 (coffee) + 2 (lunch) + 1 (groceries) + 1 (hardware store)

Number of purchases = 8

Average cost = Total cost / Number of purchases

Average cost = $112.56 / 8

Average cost = $14.07

Therefore, the average cost of the items Roy bought is $14.07.

The 80% confidence interval for the average water usage of 249 families is (15.42, 16.18) gallons.

To construct an 80% confidence interval for the average water usage of 249 families, we can use the following formula:

CI = X ± t(α/2, df) * (s/√n)

where X is the sample mean (15.8 gallons), t(α/2, df) is the critical t-value from the t-distribution table at the desired confidence level and degrees of freedom (247 in this case), s is the population standard deviation (1.7 gallons), and n is the sample size (249).

At an 80% confidence level and 247 degrees of freedom, the critical t-value is approximately 1.296.

Plugging in the values, we get:

CI = 15.8 ± 1.296 * (1.7/√249)

CI = 15.8 ± 0.38

Therefore, the 80% confidence interval for the average water usage of 249 families is (15.42, 16.18) gallons.

This means that we are 80% confident that the true average water usage of the population lies within this interval.

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The inches of water could Rafael have added is less than 3/8 inches

Given that

So, the inequality expression is

Initial water level - The decreased amount + Amount of water added by Rafael < The actual water level

Represent the Amount of water added by Rafael with x

So, we have

11 1/2 - 7/8 + x < 11

So, we have

92 - 7 + 8x < 88

When the like terms are evaluated, we have

8x < 88 + 7 - 92

Evaluate

8x <3

This gives

x < 3/8

Hence, the amount added by Rafael is less than 3/8 inches

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Answer:

7:4 that would be the ratio

Yes, repeating two simple arithmetic operations will eventually transform every positive integer into 1.

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

f(n) = n/2 if n%2 = 0

f(n) = 3n + 1 if n%2 = 1

The above definition means that

f(n) = n/2 if n is even

f(n) = 3n + 1 if n is odd

To check if repeating operations would transform to 1, we can set n = 10

and then evaluate the function values

So, we have

f(10) = 10/2 = 5

f(5) = 3(5) + 1 = 16

f(16) = 16/2 = 8

f(8) = 8/2 = 4

f(4) = 4/2 = 2

f(2) = 2/2 = 1

See that the end result of the operations is 1

Hence, repeating two operations will transform every positive integer into 1

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Answer:

25%

Step-by-step explanation:

The box and whisker plot is broken up into 4 quartiles.  Each quartile is 25% of the date.  The data range in in the third quartile.

Helping in the name of Jesus.

Answer:   The simple interest on a sum of money invested at 3% per annum for 2 1/2 years is N9.225, and the principal is N3,000.

Step-by-step explanation:

We can use the formula for simple interest:

I = P * r * t

where I is the interest, P is the principal, r is the interest rate, and t is the time in years.

Given that the interest is N123, the interest rate is 3% per annum, and the time is 2 1/2 years, we can plug these values into the formula and solve for the principal:

123 = P * 0.03 * 2.5

Simplifying, we get:

123 = 0.075P

P = 123 / 0.075

P ≈ 1,640

Therefore, the principal is approximately N1,640.

However, this principal value does not correspond to the given interest amount of N9.225. To obtain the correct principal, we need to use the correct interest amount:

We can convert the time of 2 1/2 years to 5/2 years, and use the formula for simple interest:

I = P * r * t

where I is the interest, P is the principal, r is the interest rate, and t is the time in years.

Given that the interest is N9.225, the interest rate is 3% per annum, and the time is 5/2 years, we can plug these values into the formula and solve for the principal:

9.225 = P * 0.03 * 5/2

Simplifying, we get:

9.225 = 0.0375P

P = 9.225 / 0.0375

P = 246

Therefore, the principal is N246.

Answer:

d

Step-by-step explanation:

The first term is   'a'

  then for the second term    a r

      third  term =    a r^(n-1)

         etc  

This is a word problem, and the cost of one house in rubles is 1200, as the five sisters shared their father's inheritance equally.

If the three oldest sisters each paid 800 rubles, then they collectively paid 2400 rubles (800 rubles x 3 sisters) to the two younger sisters

Each of the the two younger sisters will have a total of 1200 rubles (2400 rubles/2 younger sisters), as they divided the money among themselves.

In conclusion, since the five sisters shared their father's inheritance equally, then 1200 rubles is the cost of one house, which solves the word problem

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There is no proportional relationship between x and y.

No, all ratios yx are not equivalent.

We have,

"Two or more number or variable are said to be in proportion if the ratio between them are equivalent to each other."

According to the question,

Given x : 8 , 10, 12, 14

        y : 5, 7, 9, 11

Ratio between different values of x and y are not equivalent

( 8 /5) ≠ (10 / 7)≠ (12 /9) ≠(14 /11)

We conclude there is no proportional relationship between x and y.

Hence, Option(1) is the correct answer.

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complete question:

Is there a proportional relationship between x and y? Explain.

x 8 10 12 14

y 5 7 9 11

1. No; all ratios yx are not equivalent.

2.Yes; each x value and y value increases by 2.

3.Yes; the difference between each x value and y value is 3.

4.No; all ratios xy are greater than 1.

According to the given function.

There were approximately 24.9 million international Netflix subscriptions in the year 2012.

There were approximately 167.7 million international Netflix subscriptions in the year 2020,

We have,

a.

Since the function N(x) gives the number of international Netflix subscriptions x years after 2012, to find the number of Netflix subscriptions in the year 2012,

We need to set x = 0.

So,

N(0) = 24.9(1.736)^0

       = 24.9 million

b.

To estimate the number of Netflix subscriptions in the year 2020, we need to set x = 8 (since 2020 is 8 years after 2012).

N(8) = 24.9(1.736)^8

       =  167.7 million

Thus,

According to the given function.

There were approximately 24.9 million international Netflix subscriptions in the year 2012.

There were approximately 167.7 million international Netflix subscriptions in the year 2020,

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