A rectangle Is removed from a right triangle to create the shaded region shown below.if

Answer:

Midpoint = (-1, -1/2)

Step-by-step explanation:

We can find the midpoint using the midpoint formula which is given by:

M = (x1 + x2) / 2, (y1 + y2) / 2, where

Thus, we can plug in (-9, -1) for (x1, y1) point and (7, 0) for (x2, y2) in the midpoint formula:

M = (-9 + 7) / 2, (-1 + 0) / 2

M = (-2) / 2, (-1) / 2

M = -1, -1/2

Thus, (-1, -1/2) is the midpoint of the line segment with endpoints (-9, -1) and (7, 0).

Answer:

[tex]12000 ( {1 + \frac{.04}{12}) }^{12t} = 17000[/tex]

[tex] {( \frac{301}{300}) }^{12t} = \frac{17}{12} [/tex]

[tex]12t (ln(301) - ln(300) ) = ln(17) - ln(12) [/tex]

[tex]t = \frac{ ln(17) - ln(12) }{12( ln(301) - ln(300) )} = 8.72 \: years[/tex]

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The range of the rational function is the set of all real numbers larger than 60 but less than 560, therefore;

Range; [60, 560]

A rational function, f(x) is a function that can be expressed in the form f(x) = p(x)/q(x), where the functions p(x) and q(x) are polynomial functions.

The initial fee charged by the cell phone company = $500

The monthly charge after purchase = $60

The total cost of owning the cell phone = $500 + $60·t

The average monthly cost of owning a cell phone is therefore;

C(t) = (500 + 60·t)/t

The range of the function is the set of all possible values o C(t), which can be frond from the limit of the function, as follows;

[tex]\lim\limits_{x\to\infty}C(t) = \lim\limits_{x\to\infty}\frac{500 + 60\cdot t}{t} = \lim\limits_{x\to\infty}(\frac{500 }{t} + 60)[/tex] = 60

The limit of the average monthly cost indicates that the range of the function approaches $60 as t approaches infinity.

When t = 1, we get; C(1) = (500 + 60 × 1)/1 = 560

The range of the function is therefore, the set of all real numbers, larger than $60 but less than $560

The range of the function is therefore; [60, 560]

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Answer: 180 gallons needed

Step-by-step explanation:

Zykeith,

Assume x gallons of 50% antifreeze is needed

Final mixture is x + 60 gallons

Amount of antifreeze in mixture is 0.4*(x+60)

Amount of antifreeze added is .5x + .1*60 = .5x + 6

so .5x + 6 = .4(x + 60)

.5x -.4x = 24 -6

.1x = 18

x = 180

Let x be the number of gallons of the 50% antifreeze solution needed. We know that the resulting mixture will be 70 + x gallons. To get a 40% antifreeze mixture, we can set up the following equation:

[tex]{\implies 0.5x + 0.1(70) = 0.4(70 + x)}[/tex]

Simplifying the equation:

[tex]\qquad\implies 0.5x + 7 = 28 + 0.4x[/tex]

[tex]\qquad\quad\implies 0.1x = 21[/tex]

[tex]\qquad\qquad\implies \bold{x = 210}[/tex]

[tex]\therefore[/tex] We need 210 gallons of the 50% antifreeze solution to mix with 70 gallons of 10% antifreeze to get a mixture that is 40% antifreeze.

[tex]\blue{\overline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}}[/tex]

The appropriate shape to model each section of the tower are the cone and the cylinder.

The approximate surface area of each shape would be =

For cone = 1,041.27m²

For cylinder = 3,543.72m².

The first shape is a cone and the formula for the surface area = A = πr(r+√h²+r²)

where;

Radius = 24/2 = 12

height = 10m

Area = 1,041.27m²

For cylinder:

A = 2πrh+2πr²

where:

r = 12m

h = 35m

A = 3,543.72m²

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Answer:

C. (n+5)(n-5)

Step-by-step explanation:

Select the expression that is equivalent to (n²-25)

Let's check each option. A and B are wrong, so we only check C & D.

C. (n + 5) (n - 5)

n² - 5n + 5n - 25

n² - 25

D. (n - 5)²

(n - 5) (n - 5)

n² - 5n - 5n + 25

n² - 10n + 25

So, the correct answer is C. (n+5)(n-5)

Answer: The prime factorization of 15 is 3x5.

a. The initial velocity of the marble is 0 cm/s.

b. The marble is rolling at a speed of 80 cm/s at 4 seconds.

c. The velocity is 50 cm/s at approximately t = 2√2 - 1 seconds.

d. The marble is rolling at a speed of 90 cm/s when it is 90 cm from its starting point at t = 9 seconds.

e. lim s(t) as t approaches infinity is 100 cm and lim v(t) as t approaches infinity is 0 cm/s; the model may not be valid for large values of t as it assumes the marble is rolling up an inclined plane without considering other factors such as friction.

a. To find the initial velocity of the marble, we need to calculate the limit of the function s(t) as t approaches 0:

lim (t->0) s(t) = lim (t->0) (100t / (t + 1))

By substituting 0 into the expression, we get:

lim (t->0) (0 / (0 + 1)) = 0 / 1 = 0.

Therefore, the initial velocity of the marble is 0 cm/s.

b. To find the speed of the marble at time 4 seconds, we substitute t = 4 into the expression for s(t):

s(4) = 100(4) / (4 + 1) = 400 / 5 = 80 cm/s

The marble is rolling at a speed of 80 cm/s at 4 seconds.

c. To find the time at which the velocity is 50 cm/s, we set s'(t) (the derivative of s(t)) equal to 50 and solve for t:

s'(t) = 50

[tex](100 / (t + 1))^2 = 50[/tex]

100 / (t + 1) = ±√50

100 = ±√50(t + 1)

±√50(t + 1) = 100

t + 1 = 100 / ±√50

t + 1 = ±2√2

Since time cannot be negative, we take t + 1 = 2√2:

t = 2√2 - 1

The velocity is 50 cm/s at approximately t = 2√2 - 1 seconds.

d. To find the speed of the marble when it is 90 cm from its starting point, we need to solve the equation s(t) = 90 for t:

100t / (t + 1) = 90

100t = 90(t + 1)

100t = 90t + 90

10t = 90

t = 9

The marble is rolling at a speed of 90 cm/s when it is 90 cm from its starting point, which occurs at t = 9 seconds.

e. The limit of s(t) as t approaches infinity (lim s(t) as t->∞) is calculated by considering the dominant term in the numerator and denominator:

lim (t->∞) (100t / (t + 1))

≈ lim (t->∞) (100t / t)

= lim (t->∞) 100

= 100

Therefore, lim s(t) as t approaches infinity is 100 cm.

Similarly, the limit of v(t) (velocity) as t approaches infinity (lim v(t) as t->∞) can be found by taking the derivative of s(t) and evaluating the limit:

[tex]v(t) = s'(t) = 100 / (t + 1)^2[/tex]

lim (t->∞) v(t) = lim (t->∞) (100 / [tex](t + 1)^2)[/tex]

≈ lim (t->∞)[tex](100 / t^2)[/tex]

= lim (t->∞) [tex](100 / t^2)[/tex]

= 0.

The limit of v(t) as t approaches infinity is 0 cm/s.

As for the validity of the model for large values of t, it is important to note that the given model assumes that the marble is rolling up an inclined plane.

However, without further information about the nature of the inclined plane (e.g., its slope, frictional forces), it is difficult to determine the accuracy.

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Answer$535.50

Step-by-step explanation:

15% is equal to .15

So, multiply 600.00x .15=90

                   600.00 - 90.0=510.
                   510. 00x .05=25.50

                   510.00+25.50=535.50

                   Your answer is $535.5

The midpoint of AB is M(1,2). If the coordinates of A are (8,-4), The coordinates of point B are (-6, 8).

To find the coordinates of point B, we can use the midpoint formula, which states that the midpoint between two points (x₁, y₁) and (x₂, y₂) is given by:

Midpoint = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

We are given the midpoint M(1, 2) and the coordinates of point A (8, -4). Let's denote the coordinates of point B as (x, y).

Using the midpoint formula, we can set up the following equations:

1 = (8 + x) / 2 (equation 1)

2 = (-4 + y) / 2 (equation 2)

Let's solve these equations to find the values of x and y.

From equation 1:

2 = 8 + x

x = 2 - 8

x = -6

From equation 2:

4 = -4 + y

y = 4 + 4

y = 8

Therefore, the coordinates of point B are (-6, 8).

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Answer:

B = (-6, 8)

Step-by-step explanation:

To find the coordinates of B, given the coordinates of A and the coordinates of the midpoint of AB, use the Midpoint formula.

[tex]\boxed{\begin{minipage}{7.4 cm}\underline{Midpoint formula}\\\\$M(x,y)=\left(\dfrac{x_2+x_1}{2},\dfrac{y_2+y_1}{2}\right)$\\\\\\where $(x_1,y_1)$ and $(x_2,y_2)$ are the endpoints.\\\end{minipage}}[/tex]

Let A = (x₁, y₁) = (8, -4)

Let B = (x₂, y₂)

Given M is (1, 2), substitute the points into the midpoint formula:

[tex](1,2)=\left(\dfrac{x_2+8}{2},\dfrac{y_2-4}{2}\right)[/tex]

[tex](1,2)=\left(\dfrac{x_2}{2}+4,\dfrac{y_2}{2}-2\right)[/tex]

Equate the x-coordinates and solve for x₂:

[tex]\begin{aligned}1&=\dfrac{x_2}{2}+4\\\\1-4&=\dfrac{x_2}{2}+4-4\\\\-3&=\dfrac{x_2}{2}\\\\-3\cdot 2&=\dfrac{x_2}{2}\cdot 2\\\\-6&=x_2\\\\x_2&=-6\end{aligned}[/tex]

Equate the y-coordinates and solve for y₂:

[tex]\begin{aligned}2&=\dfrac{y_2}{2}-2\\\\2+2&=\dfrac{y_2}{2}-2+2\\\\4&=\dfrac{y_2}{2}\\\\4\cdot 2&=\dfrac{y_2}{2}\cdot 2\\\\8&=y_2\\\\y_2&=8\end{aligned}[/tex]

Therefore, the coordinates of B are (-6, 8).

Answer:

-6 and 7.

Step-by-step explanation:

If we have a function called g, and we know that it has two factors: (x - 7) and (x + 6), then we can find the values of x that make g equal to zero. We call those values the "zeros" of the function g. To find the zeros, we just need to solve the equation (x - 7)(x + 6) = 0. The answer is that the zeros of g are -6 and 7.

The range of the inverse of the function is [2, ∝)

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

f(x) = √x - 2

Set the radicand greater tahn or equal to 0

So, we have

x - 2 ≥ 0

When evaluated, we have

x ≥ 2

This means that

[2, ∝)

Hence, the range of the inverse of the function is [2, ∝)

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A triangle JKL has vertices at J(−1, −5), K(−2, −2), and L(2, −4) and J′ at (−3, −5). The translation direction is 2 units to the left. The number of units of the image of triangle JKL is 2 units to the left only.

Given that a triangle JKL has vertices at J(−1, −5), K(−2, −2), and L(2, −4) and J′ at (−3, −5). We have to determine the translation direction and the number of units of the image of triangle JKL. Let's first find the translation direction to determine the image of triangle JKL.

Seeing the position of J and J', we can determine that the translation was made in the left direction because J has moved from the point (-1,-5) to (-3,-5). Thus, the translation direction is 2 units to the left. Now, let's calculate the number of units of the image of triangle JKL.

Let's draw a rough sketch of the triangle JKL and locate its vertices J(-1,-5), K(-2,-2), and L(2,-4).To find the number of units of the image of triangle JKL, we need to find the horizontal and vertical distances between the vertices of the original triangle and its image.

We can use the horizontal distance between J and J′ as a reference to calculate the remaining distances. J has moved 2 units to the left, so the horizontal distance between J and J′ is 2. Now, let's calculate the vertical distance between J and J′. The coordinates of J and J′ are (-1,-5) and (-3,-5), respectively.

The difference between the y-coordinates of J and J′ is 0, which means that J and J′ are on the same horizontal line. Therefore, the vertical distance between J and J′ is 0. Hence, the image of the triangle JKL has moved 2 units to the left and 0 units vertically. Thus, the number of units of the image of triangle JKL is 2 units to the left only.

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The statement "There is one outlier, indicating an abnormally large number of books were checked out on that day" is true based on the data set.

To determine if there are any outliers in the given data set of the number of books checked out from a library during the first two weeks of the month, we can examine the values and look for extreme values that deviate significantly from the rest of the data.

The data set is as follows:

19, 10, 15, 99, 12, 18, 15, 16, 12, 13, 18, 17, 19, 13

To identify outliers, we can use different methods, such as the interquartile range (IQR) or z-scores. Let's calculate the IQR to determine if there are any outliers:

Arrange the data set in ascending order:

10, 12, 12, 13, 13, 15, 15, 16, 17, 18, 18, 19, 19, 99

Calculate the first quartile (Q1) and third quartile (Q3):

Q1 = 13 (median of the lower half)

Q3 = 18 (median of the upper half)

Calculate the interquartile range (IQR):

IQR = Q3 - Q1 = 18 - 13 = 5

Define the lower and upper boundaries for outliers:

Lower bound = Q1 - 1.5 * IQR = 13 - 1.5 * 5 = 5.5

Upper bound = Q3 + 1.5 * IQR = 18 + 1.5 * 5 = 25.5

Based on these calculations, any value below 5.5 or above 25.5 would be considered an outlier.

Looking at the data set, we can see that there is indeed one outlier, which is the value 99. It is significantly larger than the rest of the values and falls above the upper bound.

Therefore, the statement "There is one outlier, indicating an abnormally large number of books were checked out on that day" is true based on the data set.

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The minimum amount of cholesterol is 350 units.

To minimize cholesterol intake while meeting the daily supplement requirements, we need to find the optimal amounts of foods I and II to consume. Let's denote the amount of food I as x ounces and the amount of food II as y ounces.

We have the following constraints:
- The total supplement of the two foods must be at most 22 ounces: x + y ≤ 22
- The daily vitamin C requirement is at least 380 units: 76x + 38y ≥ 380
- The daily vitamin E requirement is at least 170 units: 10x + 20y ≥ 170

To minimize cholesterol intake, we need to minimize the amount of cholesterol from both foods. Food I contains 10 units of cholesterol per ounce, so the cholesterol from food I is 10x. Food II contains 16 units of cholesterol per ounce, so the cholesterol from food II is 16y. Therefore, the total cholesterol is 10x + 16y.

Now, let's solve this problem using linear programming:

Step 1: Rewrite the constraints in terms of x and y:
x + y ≤ 22
76x + 38y ≥ 380
10x + 20y ≥ 170

Step 2: Graph the feasible region determined by these constraints.

Step 3: Identify the corner points of the feasible region.

Step 4: Substitute the corner points into the objective function 10x + 16y and find the minimum value.

After performing these steps, we find that the minimum amount of cholesterol is 350 units.

The values for x and y that correspond to the minimum cholesterol intake may vary, so it is important to verify the optimal solution by substituting the values into the constraints to ensure they satisfy all the requirements.

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Answer:

x = 4 , ∠ AXC = 150°

Step-by-step explanation:

∠ 1 and ∠ 2 form the angle AXC , that is

∠ AXC = ∠ 1 + ∠ 2 , then

6(6x + 1) = 102 + 10x + 8

36x + 6 = 10x + 110 ( subtract 10x from both sides )

26x + 6 = 110 ( subtract 6 from both sides )

26x = 104 ( divide both sides by 26 )

x = 4

Then by substituting x = 4

∠ AXC = 6(6x + 1) = 36x + 6 = 36(4) + 6 = 144 + 6 = 150°

The correct matches are:

-1: 0

-2: 0

X: 14/3

0: 0

6: Not used

= 2² + 2x - 5: Not used

To match the representations with their respective average rates of change, we need to calculate the average rate of change for each function over the interval (0, 3) and compare it to the given values.

Let's calculate the average rate of change for each function:

Function: 2² + 2x - 5

To find the average rate of change, we need to calculate the difference in function values divided by the difference in x-values:

Average rate of change = (f(3) - f(0)) / (3 - 0)

Average rate of change = ((2² + 2(3) - 5) - (2² + 2(0) - 5)) / 3

Average rate of change = (13 - (-1)) / 3

Average rate of change = 14 / 3

Match: X = 14/3

Function: -1

Since the function is constant, the average rate of change is 0.

Match: 0

Function: 2

Since the function is constant, the average rate of change is 0.

Match: 0

Function: 3

Since the function is constant, the average rate of change is 0.

Match: 0

Function: -2

Since the function is constant, the average rate of change is 0.

Match: 0

Function: -3

Since the function is constant, the average rate of change is 0.

Match: 0

Therefore, the correct matches are:

-1: 0

-2: 0

X: 14/3

0: 0

6: Not used

= 2² + 2x - 5: Not used

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When x = 3, the expression 2x - 50 evaluates to -44.

To demonstrate an example using the expression 2(x + 5) - 5 × 12, let's simplify it step by step:

Start with the given expression.

2(x + 5) - 5 × 12

Apply the distributive property.

2x + 2(5) - 5 × 12

Simplify within parentheses and perform multiplication.

2x + 10 - 60

Combine like terms.

2x - 50

The simplified form of the expression 2(x + 5) - 5 × 12 is 2x - 50.

Let's consider an example for substituting a value for the variable x:

Suppose we want to evaluate the expression when x = 3. We substitute x = 3 into the simplified expression:

2(3) - 50

Now, perform the calculations:

6 - 50

The result is -44.

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Question

evaluate the expression 2(x+5)-5 x 12.

The line of reflection for the image of triangle DEF is the y-axis.

To determine the line of reflection for the image of triangle DEF, we need to find the axis along which the reflection occurred. We can do this by examining the coordinates of the corresponding points before and after the reflection.

Given that D' is located at (3, 5) after the reflection, we can compare the x-coordinates of D and D'. The x-coordinate of D is -3, and the x-coordinate of D' is 3. We notice that there is a change in sign, indicating a reflection across the y-axis.

Therefore, the line of reflection for the image of triangle DEF is the y-axis.

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The expression that is equivalent to x√6 is option C, 62/37/3.The correct choice is C. 62/37/3.

To find an expression that is equivalent to √(6x²), we need to simplify the square root.

Using the properties of square roots, we know that the square root of a product is equal to the product of the square roots. Therefore, we can simplify the expression as follows:

√(6x²) = √6 * √(x²)

The square root of x² is simply x, and the square root of 6 cannot be simplified further. Therefore, the expression can be simplified as:

√(6x²) = x√6

Among the given options, the expression that is equivalent to x√6 is option C, 62/37/3.

Therefore, the correct choice is C. 62/37/3.

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a.The Industry Average Salary is $55,680. b.The Firm A's Average Salary is $51,718.40 .c. Firm B's average salary is approximately 112.27% of Firm A's average salary.

a. To determine the industry average salary, we can use the information that the industry average salary is 96% of Firm B's average salary. Firm B's average salary is $58,000. Therefore, we can calculate the industry average salary as follows:

Industry Average Salary = 96% of Firm B's Average Salary

= 0.96 * $58,000

= $55,680

b. Firm A's average salary is stated as 93% of the industry average salary. To calculate Firm A's average salary, we can multiply the industry average salary by 93%:

Firm A's Average Salary = 93% of Industry Average Salary

= 0.93 * $55,680

= $51,718.40

c. To express Firm B's average salary as a percentage of Firm A's average salary, we can divide Firm B's average salary by Firm A's average salary and multiply by 100:

Percentage = (Firm B's Average Salary / Firm A's Average Salary) * 100

= ($58,000 / $51,718.40) * 100

≈ 112.27%

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The critical value tₕ for a confidence level of 0.99 and a sample size of 22 is approximately 2.831.

To find the critical value tₕ for a given confidence level and sample size, we can use a t-distribution table or a statistical software.

For a confidence level of 0.99, we need to determine the critical value tₕ such that the area under the t-distribution curve to the right of tₕ is equal to (1 - c) / 2. In this case, (1 - c) / 2 = (1 - 0.99) / 2 = 0.005.

Since the sample size is 22, we have n - 1 degrees of freedom, which is 22 - 1 = 21. We need to find the critical value tₕ with 21 degrees of freedom that corresponds to an area of 0.005 in the upper tail of the t-distribution.

Using a t-distribution table or a statistical software, we find that the critical value tₕ is approximately 2.831.

Therefore, the critical value tₕ for a confidence level of 0.99 and a sample size of 22 is approximately 2.831.

It is important to note that the critical value may vary slightly depending on the specific t-distribution table or statistical software used. However, the value obtained above is a reasonable approximation based on commonly used tables or software.

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The simplified form of the expression 1200 × 1260 ÷ 800 in standard form is 1890.

To simplify the given expression, we perform the multiplication and division operations according to the order of operations (PEMDAS/BODMAS).

First, we perform the multiplication: 1200 × 1260 = 1,512,000.

Next, we perform the division: 1,512,000 ÷ 800 = 1890.

The result, 1890, is in standard form.

In standard form, a number is expressed as a product of a number between 1 and 10 (inclusive) and a power of 10. In this case, 1890 is already in the appropriate format and does not require any further modification.

Therefore, the simplified form of the expression 1200 × 1260 ÷ 800 is 1890 in standard form.

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Answer:

380 seconds

Step-by-step explanation:

Convert 6 minutes to seconds by multiplying 6 times 60, because there are 60 seconds per minute.
6 x 60 = 360

Now add the 20 seconds.
360 + 20 = 380

6 minutes and 20 seconds are equal to 380 seconds.

In the class , each child needs to pay (n * p) / y shillings to cover the cost of the lost books.

In the given scenario, there is a class of "y" children, and "n" mathematics books were lost, each costing "p" shillings. The teacher decides to distribute the cost of the lost books equally among all the children.

To find out how much money each child needs to pay, we need to divide the total cost of the lost books by the number of children in the class.

The total cost of the lost books can be calculated by multiplying the number of lost books ("n") by the cost of each book ("p").

Total cost = n * p

To distribute the cost equally among all the children, we divide the total cost by the number of children ("y"):

Cost per child = Total cost / Number of children

Substituting the values, we get:

Cost per child = (n * p) / y

Therefore, each child in the class needs to pay (n * p) / y shillings to cover the cost of the lost books.

It's important to note that this calculation assumes that the total cost is evenly distributed among all the children, regardless of whether they were responsible for losing the books or not.

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Answer:

B

Step-by-step explanation:

Yes, because for each input there is exactly one output. You can have two of the same x values but you cannot have 2 of the same y values. if you have two of the same y values, it is not a function as it doesn't pass the vertical line test.

i. The payback period for the project is 6 years.

ii. The accounting rate of return (ARR) using the average investment method is 21.18%.

iii. The net present value (NPV) of the project is -165,143.

iv. The internal rate of return (IRR) of the project is approximately 19.61%.

i. The payback period for the project:

To calculate the payback period, we need to determine how long it takes for the cumulative net cash flows to equal or exceed the initial investment of 350 million + 150 million.

Year 1: 70,000, Year 2: 70,000, Year 3: 80,000, Year 4: 100,000, Year 5: 100,000, Year 6: 120,000.

Cumulative Cash Flow:

Year 1: 70,000

Year 2: 70,000 + 70,000 = 140,000

Year 3: 140,000 + 80,000 = 220,000

Year 4: 220,000 + 100,000 = 320,000

Year 5: 320,000 + 100,000 = 420,000

Year 6: 420,000 + 120,000 = 540,000.

The cumulative cash flows exceed the initial investment of 500 million (350 million + 150 million) in Year 6.

So, the payback period for the project is 6 years.

ii. The accounting rate of return (ARR) using the average investment method:

ARR = Average Annual Profit / Average Investment

Average Annual Profit = Sum of Net Cash Flows / Number of Years

Average Annual Profit = (70,000 + 70,000 + 80,000 + 100,000 + 100,000 + 120,000) / 6

Average Annual Profit = 540,000 / 6

Average Annual Profit = 90,000

Average Investment = (Initial Investment + Residual Value) / 2

Average Investment = (500 million + 350 million) / 2

Average Investment = 425 million.

ARR = 90,000 / 425,000 = 0.2118 or 21.18%

iii. The net present value (NPV) of the project:

To calculate NPV, we discount each cash flow to its present value using the cost of capital of 12%.

NPV = (Net Cash Flow1 / [tex](1 + r)^1)[/tex]  + (Net Cash Flow2 / [tex](1 + r)^2)[/tex] + ... + (Net Cash Flow6 / (1 + r)^6) - Initial Investment.

[tex]NPV = (70,000 / (1 + 0.12)^1) + (70,000 / (1 + 0.12)^2) + (80,000 / (1 + 0.12)^3) + (100,000 / (1 + 0.12)^4) + (100,000 / (1 + 0.12)^5) + (120,000 / (1 + 0.12)^6) -[/tex] (350 million + 150 million)

Calculating each term and summing them up:

NPV = 54,017 + 48,234 + 54,497 + 62,313 + 55,631 + 60,165 - 500 million

NPV = -165,143

Therefore, the net present value (NPV) of the project is -165,143.

iv. The internal rate of return (IRR) of the project:

To calculate the IRR, we find the discount rate that makes the NPV equal to zero. Using a financial calculator or Excel, we can determine that the IRR for this project is approximately 19.61%.

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The third term (a3) of the sequence is -8, the fourth term (a4) is 35, and the fifth term (a5) is -183. These values are obtained by applying the given formula recursively and substituting the previous terms accordingly. The calculations follow a specific pattern and are derived using the provided formula.

The sequence is defined by the following formula:
a1 = 1, a2 = 3,
and
an = (-1)nan - 1 + an - 2 for n ≥ 3.
To find the third term (a3), we substitute n = 3 into the formula:
a3 = (-1)(3)(a3 - 1) + a3 - 2.
Next, we simplify the equation:
a3 = -3(a2) + a1.
Since we know a1 = 1 and a2 = 3, we substitute these values into the equation:
a3 = -3(3) + 1.
Simplifying further:
a3 = -9 + 1.
Therefore, the third term (a3) is equal to -8.
To find the fourth term (a4), we substitute n = 4 into the formula:
a4 = (-1)(4)(a4 - 1) + a4 - 2.
Simplifying the equation:
a4 = -4(a3) + a2.
Since we know a2 = 3 and a3 = -8, we substitute these values into the equation:
a4 = -4(-8) + 3.
Simplifying further:
a4 = 32 + 3.
Therefore, the fourth term (a4) is equal to 35.
To find the fifth term (a5), we substitute n = 5 into the formula:
a5 = (-1)(5)(a5 - 1) + a5 - 2.

Simplifying the equation:
a5 = -5(a4) + a3.
Since we know a4 = 35 and a3 = -8, we substitute these values into the equation:
a5 = -5(35) + (-8).
Simplifying further:
a5 = -175 - 8.
Therefore, the fifth term (a5) is equal to -183.
In summary, the third term (a3) is -8, the fourth term (a4) is 35, and the fifth term (a5) is -183.

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