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Problem 10

# 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.

Expert verified
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.
See the step by step solution

## 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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