The weight (in pounds) and height (in inches) for a child were measured every few months over a two-year period. The results are given in the table.



A 2-column table with 9 rows. Column 1 is labeled Weight (x) with entries 8, 12, 18, 24, 30, 32, 35, 37, 40. Column 2 is labeled Height (y) with entries 22, 23, 26, 30, 32, 33, 35, 36, 38.



Using technology, what is the correlation coefficient?



–0. 997


–0. 503


0. 503


0. 997

To find the derivative (f'(x)) of the function f(x) = 2x^4 In x, we will need to use the product rule and the chain rule of differentiation.

Using the product rule, we have:

f'(x) = [2(In x)](4x^3) + [2x^4](1/x)

Simplifying this expression, we get:

f'(x) = 8x^3 In x + 2x^3

Therefore, the derivative of f(x) is f'(x) = 8x^3 In x + 2x^3.
Hi! To find the derivative f'(x) of the function f(x) = 2x^4 * ln(x), we'll use the product rule. The product rule states that if you have a function h(x) = u(x)v(x), then h'(x) = u'(x)v(x) + u(x)v'(x). In this case, u(x) = 2x^4 and v(x) = ln(x).

First, find the derivatives of u(x) and v(x):
u'(x) = d(2x^4)/dx = 8x^3
v'(x) = d(ln(x))/dx = 1/x

Now, apply the product rule:
f'(x) = u'(x)v(x) + u(x)v'(x)
f'(x) = (8x^3)(ln(x)) + (2x^4)(1/x)

Simplify the expression:
f'(x) = 8x^3 * ln(x) + 2x^3

This is the derivative of the given function.

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Step-by-step explanation:

the

answer

of

this

question

is

9 \sqrt{2}

Do you like this (^__^) ??

The graph of the function will touch the x-axis at x = -5, but not cross it, and the behavior of the graph near x = -5 will be determined by the degree of the zero (which is 3 in this case).

The polynomial function that represents the volume of a sphere with radius x+5 is given by:

[tex]V(x) = (4/3)\pi (x+5)^3[/tex]

To find the multiplicity of the zero, we need to factor out the (x+5) term from the polynomial:

V(x) = (4/3)π(x+5)(x+5)(x+5)

We can see that the zero is x = -5, and it has a multiplicity of 3, since there are three factors of (x+5) in the polynomial.

This means that the graph of the function will touch the x-axis at x = -5, but not cross it, and the behavior of the graph near x = -5 will be determined by the degree of the zero (which is 3 in this case).

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In general, it is best to determine the equation of the line of best fit exactly rather than relying on estimation by inspection. This is because an exact equation allows for more precise predictions and calculations.

Estimation by inspection can be useful in situations where the data is relatively simple and a rough estimate is sufficient. However, in more complex datasets, it is important to use statistical methods to determine the line of best fit accurately.

For example, in some cases, it may be more important to prioritize the accuracy of the slope of the line over the accuracy of the intercept. In such cases, certain methods, such as minimizing the sum of the squares of the vertical deviations, may be more appropriate than others.

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Nosaira's work is incorrect. Her mistake is in the step where she simplifies the expression 2x+1 on the left side of the equation by multiplying it with 3. She distributed the 3 only to the 2x term, but forgot to distribute it to the 1 term as well.

So, the correct expression on the left side should be 6x+3 instead of 6x+1. This mistake leads to the wrong equation 6x+3=2x^2-1, and when she tries to solve for x, she ends up with the equation 4x=0, which only has one solution, x=0.

Therefore, Nosaira's interpretation of the answer as having no solution is incorrect. The original equation actually does have a solution, which is x=1/2. If we correct the mistake in her work, we can see that the equation becomes [tex]6x+3=2x^2-1[/tex], which simplifies to [tex]2x^2-6x-4=0[/tex]. We can then factor out 2 to get [tex]x^2-3x-2=0[/tex], which can be factored further into (x-2)(x+1)=0. Therefore, the solutions are x=2 and x=-1, but we need to reject the negative solution as it does not satisfy the original equation.

In conclusion, Nosaira made a mistake in distributing the coefficient 3, which led to an incorrect equation and an incorrect interpretation of the answer. It is important to be careful and check our work, especially when dealing with algebraic expressions and equations.

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Over (-∞, 0) interval is Store A's profit greater than Store B's.

To determine the interval over which Store A's profit is greater than Store B's, we need to solve the inequality:

f(x) > g(x)

Substituting the given profit functions, we have:

2x > 83x

Simplifying this inequality, we can subtract 83x from both sides:

-81x > 0

Dividing both sides by -81 (and reversing the inequality because we are dividing by a negative number), we get:

x < 0

Therefore, Store A's profit is greater than Store B's for all values of x less than 0. In interval notation, we can write:

(-∞, 0)

So the interval over which Store A's profit is greater than Store B's is the open interval from negative infinity to 0.

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The length of the intercepted arc on the unit circle is equal to the radius of the circle times the angle in radians. In this case, since the unit circle has a radius of 1, the length of the intercepted arc is simply equal to the angle in radians.

So, the length of the intercepted arc for an angle of 1.5 radians is 1.5 units (since the angle is given in radians, not degrees).

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After calculating the distance, Ranjan has driven 358.55 miles so far.

To solve this problem, we need to use the formula:

distance = fuel efficiency x fuel consumed

Let's start by calculating the total fuel consumed. At the first stop, Ranjan adds 6.7 gallons of gas, which means he consumed 6.7 gallons of gas since his tank was full at the beginning of the trip. At the second stop, he adds 3.4 gallons of gas, which means he consumed 3.4 gallons of gas between the first and second stops. Therefore, the total fuel consumed is:

6.7 + 3.4 = 10.1 gallons

Now we can calculate the distance driven using the fuel efficiency of 35.5 miles per gallon:

distance = 35.5 miles/gallon x 10.1 gallons = 358.55 miles

Therefore, Ranjan has driven 358.55 miles so far.

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The value of the addition of the functions:

F(x) + g(x) = (8x² + 34x - 56)/(x²-3x-40).

To add the two rational functions F(x) and g(x), we first need to find a common denominator. In this case, the common denominator is (x+5)(x-8), since both denominators can be factored in this way.

F(x) needs to be multiplied by (x-8) on the top and bottom to get a common denominator of (x+5)(x-8), and g(x) needs to be multiplied by (x+5) on the top and bottom to get the same common denominator.

So, we have:

F(x) = (x+7)/(x+5) * (x-8)/(x-8) = (x² - x - 56)/(x² - 3x - 40)

g(x) = 7x/(x²-3x-40) * (x+5)/(x+5) = 7x(x+5)/(x+5)(x-8) = 7x(x+5)/(x²-3x-40)

Now that both functions have the same denominator, we can add them together:

F(x) + g(x) = (x² - x - 56)/(x² - 3x - 40) + 7x(x+5)/(x²-3x-40)

To simplify this expression, we need to combine the two fractions over the common denominator:

F(x) + g(x) = (x² - x - 56 + 7x² + 35x)/(x²-3x-40)

Combining like terms in the numerator:

F(x) + g(x) = (8x² + 34x - 56)/(x²-3x-40)

So, F(x) + g(x) = (8x² + 34x - 56)/(x²-3x-40).

To solve a rational function, we generally follow these steps:

In the case of F(x) and g(x), we already simplified the sum of the functions. We can see that the denominator factors as (x+5)(x-8), which means that the function is undefined at x = -5 and x = 8. These are vertical asymptotes.

To find any horizontal asymptotes, we can use the fact that the degree of the numerator is greater than or equal to the degree of the denominator. This means that there is no horizontal asymptote; instead, the function approaches infinity as x approaches infinity or negative infinity.

Finally, we can graph the function using this information and any other relevant points, such as intercepts.

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It's "D"

It's self explanatory but y gets 1/2 of whatever x is and adds 1. So if x = 2, then it'll get half of two (which is one) and add one to it, getting two.

Answer: 15 letters.

Step-by-step explanation:

When p is the number of letters being engraved:

12.5 + .4p = 14.75 + .25p

-12.5             -12.5

.4p = 2.25 + .25p

-.25p          -.25p

.15p = 2.25

/.15     /.15

p = 15

There would need to be 15 letters engraved for the cost of the trophies to be the same. Hope this helps!

Answer:

D. Lines A and D are perpendicular.

DAnswer:

Step-by-step explanation:

Answer:

10/12

Step-by-step explanation:

(3/10)x=1/4

3x=10/4

x=10/12

The answer that gives the correct transformation of P(x) to get to I(x) is option D) I(x) = 2P(x).

This means that the function I(x) is obtained by multiplying the function P(x) by 2.

To understand why this is the correct transformation, let's consider an example:

Suppose P(x) represents the number of items produced by a factory in x hours. If we want to find the number of items produced by the factory in 2x hours, we can use the transformation I(x) = 2P(x). This is because the rate of production is constant, so in twice the time, the factory will produce twice the number of items. Therefore, multiplying the function P(x) by 2 gives us the function I(x) that represents the number of items produced by the factory in 2x hours.

Option A) I(x) = P(1/2x) means that we are compressing the function P(x) horizontally, which would result in a faster rate of change. This transformation does not make sense in the context of the problem and is not the correct transformation.

Option B) I(x) = P(2x) means that we are stretching the function P(x) horizontally, which would result in a slower rate of change. This transformation also does not make sense in the context of the problem and is not the correct transformation.

Option C) I(x) = 1/2P(x) means that we are reducing the function P(x) by half, which would result in a slower rate of change. This transformation does not match the problem statement and is not the correct transformation.

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The rewritten expression of 12 using the distributive property is 3(2 + 2)

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

12 distributive property

This means that

12

Express as 6 + 6

So, we have

12 = 6 + 6

Factor out 3 from the equation

So, we have

12 = 3(2 + 2)

The above equation has been rewritten using the distributive property.

Hence, the rewritten expression using the distributive property is 3(2 + 2)

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There were 59.2 million enterprise IM accounts in 2006 and the expected number of enterprise IM accounts in 2009 was 119.89 million.

(a) To find the number of enterprise IM accounts in 2006, we need to evaluate

N(t) at t = 0: N(0) = 2.97(0)^2 + 11.32(0) + 59.2

N(0) = 0 + 0 + 59.2

N(0) = 59.2 million

So, there were 59.2 million enterprise IM accounts in 2006.

(b) To find the expected number of enterprise IM accounts in 2009, we need to evaluate

N(t) at t = 3 (since 2009 corresponds to t = 3): N(3) = 2.97(3)^2 + 11.32(3) + 59.2

N(3) = 2.97(9) + 33.96 + 59.2

N(3) = 26.73 + 33.96 + 59.2

N(3) = 119.89 million

So, the expected number of enterprise IM accounts in 2009 was 119.89 million.

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Answer:

[tex]f(x)=x^3+11x^2+43x+65[/tex]

Step-by-step explanation:

If a polynomial function has a complex zero, then the conjugate of that complex zero is also a zero of the polynomial.

So, if (-3 - 2i) is a zero of the polynomial, then its conjugate (-3 + 2i) is also a zero of the polynomial.

Therefore, the three zeros of the polynomial function are:

The zero of a polynomial f(x) is the x-value when f(x) = 0.

According to the factor theorem, if f(a) = 0 then (x - a) is a factor of the polynomial f(x).

Therefore, the polynomial function in factored form is:

[tex]\begin{aligned}f(x) &= (x - (-5))(x-(-3-2i))(x-(-3+2i))\\&= (x +5)(x+3+2i)(x+3-2i)\end{aligned}[/tex]

Expand the brackets to write the polynomial in standard form.

[tex]\begin{aligned}f(x) &=(x +5)(x+3+2i)(x+3-2i)\\&=(x+5)(x^2+3x-2xi+3x+9-6i+2ix+6i-4i^2)\\&=(x+5)(x^2+6x+9-4i^2)\\&=(x+5)(x^2+6x+9-4(-1))\\&=(x+5)(x^2+6x+9+4)\\&=(x+5)(x^2+6x+13)\\&=x^3+6x^2+13x+5x^2+30x+65\\&=x^3+11x^2+43x+65\end{aligned}[/tex]

Therefore, the polynomial function of least degree with integral coefficients that has the given zeros -5 and (-3 - 2i) is:

[tex]f(x)=x^3+11x^2+43x+65[/tex]

f(x)=x3+11x²+43x+65

Answer:

The correct answer is ** x(y+12) or f.

We can simplify the expression 1/5x(5y+60) by multiplying the factors in the parentheses and then dividing by 5. This gives us:

```

1/5x(5y+60) = 1/5 * 5xy + 1/5 * 60x = x(y+12)

```

The basal metabolic rate for a 180-kilogram lion is approximately 766.4 Calories per day.

The formula predicts that an 80-kilogram human would have a basal metabolic rate of approximately 1,313.9 Calories per day.

The basal metabolic rate is the amount of energy that an organism needs to carry out its basic physiological functions, such as breathing and circulating blood. In this case, Kielber's law is expressed as:

R = 73 [tex]\sqrt[4]{M^3}[/tex]

Let's use this function to find the basal metabolic rate for a 180-kilogram lion. To do this, we simply substitute M = 180 into the equation and solve for R:

R = 73 [tex]\sqrt[4]{180^3}[/tex]

R = 73 [tex]\sqrt[4]{5832}[/tex]

R ≈ 766.4

Now, let's find the formula's prediction for an 80-kilogram human. Again, we simply substitute M = 80 into the equation and solve for R:

R = 73[tex]\sqrt[4]{80^3}[/tex]

R = 73[tex]\sqrt[4]{512}[/tex]

R ≈ 1,313.9

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Complete Question:

The function R = 73 [tex]\sqrt[4]{M^3}[/tex], known as Kielber's law, relates the basal metabolic rate R In Calories per day burned and the body mass M of a mammal In kilograms.

Find the basal metabolic rate for a 180-kilogram lion. Then find the formula's prediction for an 80-kilogram human. If necessary round down to the nearest 50 Calories.

The two possible values of x are -29 and 69.

To find two possible values for x, we need to use the fact that the range of

the numbers is 63.

The range is defined as the difference between the largest and smallest

numbers in the set.

First, we can find the largest and smallest numbers in the set:

Smallest number = 6

Largest number = 34

Next, we can set up two equations to represent the range of the numbers,

using the two possible scenarios for x:

Scenario 1:

If x is the smallest number in the set, then the range is equal to [tex]34 - x.[/tex]

Scenario 2: If x is the largest number in the set, then the range is equal to

[tex]x - 6[/tex].

We can then set up two equations and solve for x in each scenario:

Scenario 1:

[tex]34 - x = 63x = 34 - 63x = -29[/tex]

Scenario 2:

[tex]x - 6 = 63x = 63 + 6x = 69[/tex]

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The area of the triangle is 20 square meters.

The formula to find the area of a triangle is A = 1/2 * base * height. In this case, the base of the triangle is 8 meters and the height is 5 meters. Therefore, the area of the triangle is A = 1/2 * 8 m * 5 m = 20 m^2.

We can also check our answer by using the formula A = (b * h) / 2, where b is the base and h is the height of the triangle. Substituting the values given in the question, we get A = (8 m * 5 m) / 2 = 20 m^2. Therefore, the area of the triangle is 20 square meters.

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Answer:

-√3/2

Step-by-step explanation:

sin(x) is equal to 1/2 when x=7π/6 or 11π/6

cos(7π/6) = -√3/2

cos(11π/6) = √3/2

In the question, it says that cos(x) is <0, which means that it has to be negative

So, the answer is -√3/2

Answer:  C

Step-by-step explanation:

Think of a unit circle

sin x = -1/2 happens at 7[tex]\pi[/tex]/6 and 11[tex]\pi[/tex]/6, 3rd and 4th quadrant

Out of those 2 quadrants cos x is negative in the 3rd quadrant

So cos x= -√3/2

The equation of the tangent plane to the surface with equation - in (9) +-3 at the point (2,1,9),  first need to find the partial derivatives of the function with respect to x, y, and z. However, the surface equation provided seems to be incorrect or incomplete. Please provide the correct surface equation in the form f(x, y, z) = constant.

Let's call the function f(x, y, z) = - in (9) +-3.
∂f/∂x = 0  (since there is no x term in the function)
∂f/∂y = 0  (since there is no y term in the function)
∂f/∂z = -3/((z-9)^2)
Now we can use the formula for the equation of the tangent plane at a point (a,b,c) on a surface z=f(x,y):
z - f(a,b) = (∂f/∂x)(a,b)(x-a) + (∂f/∂y)(a,b)(y-b)
+ (∂f/∂z)(a,b)(z-c)
Plugging in the values we have, we get:
z - (- in (9) +-3)|_(2,1) = 0(x-2) + 0(y-1) - (3/((z-9)^2))|_(2,1,9)(z-9)
Simplifying:
z + in (9) - 3 = -3(z-9)
4z = 30
z = 7.5
So the equation of the tangent plane is:
z - (- in (9) +-3)|_(2,1) = (-3/((z-9)^2))|_(2,1,9)(z-9)
z - in (9) - 3 = -3(7.5-9)
z - in (9) - 3 = 4.5
z = 12.5
Therefore, the equation of the tangent plane to the surface with equation - in (9) +-3 at the point (2,1,9) is:

z - in (9) - 3 = -3/((z-9)^2)(z-9)
z - in (9) - 3 = -3(z-9)/(z-9)^2
z - in (9) - 3 = -3/(z-9)
z = 3/(z-9) + in (9) + 3
or
3x + 3y - 4z = -27 + in (9)
To find the equation of the tangent plane to the surface with equation ln(9) +- 3 at the point (x, y, z) = (2, 1, 9), follow these steps:
1. Determine the gradient vector of the given surface at the point (2, 1, 9).
2. Use the gradient vector as the normal vector of the tangent plane.
3. Write the equation of the tangent plane using the normal vector and the given point.
However, the surface equation provided seems to be incorrect or incomplete. Please provide the correct surface equation in the form f(x, y, z) = constant, so that I can help you find the equation of the tangent plane.

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To find the velocity and acceleration 8 seconds after the rocket is launched, we need to find the first and second derivatives of the height function with respect to time.

The first derivative of h(t) gives the velocity function v(t):
v(t) = h'(t) = 3t^2 + 1.2t

Substituting t = 8 into this equation gives us the velocity of the rocket at 8 seconds after launch:
v(8) = 3(8)^2 + 1.2(8) = 204.8 feet per second

So the velocity of the rocket 8 seconds after launch is 204.8 feet per second.

The second derivative of h(t) gives the acceleration function a(t):
a(t) = h''(t) = 6t + 1.2

Substituting t = 8 into this equation gives us the acceleration of the rocket at 8 seconds after launch:
a(8) = 6(8) + 1.2 = 49.2 feet per second squared

So the acceleration of the rocket 8 seconds after launch is 49.2 feet per second squared.
To find the velocity and acceleration of the rocket 8 seconds after it is launched, we need to determine the first and second derivatives of the height function h(t) with respect to time t.

Given h(t) = t^3 + 0.6t^2, let's find its first and second derivatives:

1. Velocity (first derivative of h(t)):
v(t) = dh/dt = 3t^2 + 1.2t

2. Acceleration (second derivative of h(t)):
a(t) = d^2h/dt^2 = d(v(t))/dt = 6t + 1.2

Now, let's evaluate the velocity and acceleration at t = 8 seconds:

Velocity at t=8:
v(8) = 3(8^2) + 1.2(8) = 192 + 9.6 = 201.6 ft/s

Acceleration at t=8:
a(8) = 6(8) + 1.2 = 48 + 1.2 = 49.2 ft/s^2

So, 8 seconds after the rocket is launched, its velocity is 201.6 ft/s and its acceleration is 49.2 ft/s^2.

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Using quadratic formula, the value of x in the quadratic equation are 1.47 and -0.85

To find the value of x, we can either use quadratic formula or factorization method.

8x² - 2 - 5x = 8

Let's rewrite the equation properly

8x² - 5x - 2 - 8 = 0

8x² - 5x - 10 = 0

a = 8, b = -5, c = -10

Using quadratic formula;

-b ±[√b² - 4ac / 2a]

-(-5) ±[√(-5)² - 4(8)(-10) / 2(8)]

x = 5+ √345 / 16 or x = 5 - √345 / 16

x = 1.47 or x = -0.85

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The correct answer for the equation is  [tex]h = -2cos(\pi t) + 5[/tex] . The correct option is (1)

Given:

[tex]h= -2sin(\pi\tfrac{t}{2} )[/tex]

Examine the answer choices:

[tex]h = -2cos(\pi t) + 5[/tex]

Amplitude: |-2| = 2 (same as the given equation)

Frequency: π (same as the given equation)

Phase Shift: None (different from the given equation)

[tex]h = -2cos(\pi (t/2)) + 5[/tex]

Amplitude: |-2| = 2 (same as the given equation)

Frequency: π/2 (different from the given equation)

Phase Shift: None (different from the given equation)

[tex]h = 2cos(\pi t) + 5[/tex]

Amplitude: |2| = 2 (different from the given equation)

Frequency: π (same as the given equation)

Phase Shift: None (different from the given equation)

[tex]h = 2cos(\pi(t/2)) + 5[/tex]

Amplitude: |2| = 2 (different from the given equation)

Frequency: π/2 (different from the given equation)

Phase Shift: None (different from the given equation)

The correct equation is [tex]h = -2cos(\pi t) + 5[/tex]  .The correct option is (1).

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The rounding method used, the estimated speed of the space shuttle after it has slowed down is 10,200 miles per hour.

To estimate the speed of the space shuttle after it has slowed down, we round each number to the nearest hundred. The speed before the decrease is rounded to 17,600 miles per hour, and the decrease in speed is rounded to 7,400 miles per hour.

Next, we subtract the rounded decrease in speed from the rounded speed before. So, 17,600 - 7,400 = 10,200 miles per hour. This result represents the estimated speed of the space shuttle after it has slowed down.

Rounding to the nearest hundred is a way to approximate the values and make calculations simpler. However, it is important to note that rounding introduces some degree of error, and the actual speed after the decrease may differ slightly from the estimated value.

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One would least expect to find fossils in the surface bedrock in the Adirondack Mountains region of New York State.

This region is known for having some of the oldest rocks in North America, dating back over a billion years. These rocks were formed through volcanic activity and mountain-building processes that occurred long before the evolution of complex life forms.

As a result, the rocks in the Adirondack Mountains are generally not rich in fossils, especially those of plants and animals that evolved much later in Earth's history.

In contrast, other regions of New York State, such as the Hudson Valley and the Finger Lakes region, have rocks that are more conducive to fossil preservation. These regions were covered by shallow seas at various times in the past, allowing for the accumulation of sediment and the preservation of fossils of marine organisms.

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The initial value of the account is: 6320

Compound interest is defined as the interest you earn on interest. This can be illustrated by using basic math: if you have $100 and it earns 10% interest each year, you'll have $110 at the end of the first year.

The general formula to find compound interest is:

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

where:

A is final amount

P is initial principal balance

r is interest rate

n is number of times interest applied per time period

t is number of time periods elapsed

We are given the equation as:

f(t) = 6320(1.054)^(t)

Thus, the initial value is 6320

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