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Applied Mathematics: For the Managerial, Life, and Social Sciences

Soo T. Tan

Chapter 11: Chapter 11

Problem 10

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Camille purchased a $15-\mathrm{yr}$ franchise for a computer outlet store that is expected to generate income at the rate of $$ R(t)=400,000 $$ dollars/year. If the prevailing interest rate is $10 \% /$ year compounded continuously, find the present value of the franchise.

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The present value of the 15-year franchise for a computer outlet store, with an income rate of $400,000/year and a continuously compounded interest rate of 10%, is approximately $3,507,600.

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Step 1: Substitute the given values into the formula

We input the given values for T, r, and R(t) into the present value formula: $$ PV = \int_0^{15} 400,000e^{-0.10t}dt $$

Step 2: Integrate with respect to t

To find the present value, we need to integrate the expression with respect to t. $$ PV = 400,000 \int_0^{15} e^{-0.10t}dt $$ To integrate, use the substitution method. Let $u=-0.10t$. Thus, $du=-0.10dt$. So, we have: $$ PV = 400,000 \int_{0}^{15} e^u-\frac{1}{0.10}du $$

Step 3: Evaluate the definite integral

Now, we integrate $e^u$ and multiply by the constant term -$\frac{1}{0.10}$. $$ PV = 400,000 -\frac{1}{0.10}\left[e^u\right]_0^{15} $$ Substitute back the original value of u, which is $u=-0.10t$. $$ PV = 400,000 -\frac{1}{0.10}\left[e^{-0.10t}\right]_0^{15} $$ Evaluate the expression at the bounds (t = 0, t = 15): $$ PV = 400,000 -\frac{1}{0.10}\left[e^{-0.10 \cdot 15} - e^{-0.10 \cdot 0}\right] $$

Step 4: Simplify and calculate the present value

Simplify and calculate the expression for the present value: $$ PV = 400,000 -\frac{1}{0.10}\left[e^{-1.5} - e^0\right] \approx 400,000 -\frac{1}{0.10}\left[0.2231 - 1\right] $$ $$ PV \approx 400,000 (1 - 10 \cdot -0.7769) \approx 400,000 (1 + 7.769) \approx 400,000 \cdot 8.769 $$ $$ PV \approx 3,507,600 $$ So, the present value of the franchise is around $3,507,600.

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Chapter 11: Chapter 11

Camille purchased a \(15-\mathrm{yr}\) franchise for a computer outlet store that is expected to generate income at the rate of $$ R(t)=400,000 $$ dollars/year. If the prevailing interest rate is \(10 \% /\) year compounded continuously, find the present value of the franchise.

Short Answer

Step by step solution

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Step 1: Substitute the given values into the formula

Step 2: Integrate with respect to t

Step 3: Evaluate the definite integral

Step 4: Simplify and calculate the present value

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More chapters from the book ‘Applied Mathematics: For the Managerial, Life, and Social Sciences’

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 12

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