Gary sells his business and receives yearly payments on a sliding scale according to the function $P (t) = 120, 000 - 20, 000 t$ where $0 \leq t \leq 6$ .

If this is treated as a continuous income stream and deposited into an account that earns $4.67 %$ compounded continuously, how much will his account be worth after 6 years?

Round your answer to the nearest dollar and do not use commas in the answer blank.
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Answer

We round the final value to the nearest dollar to get $476422$ .

Steps

Step 1 ：First, we need to calculate the total amount of money Gary will receive over the 6 years by integrating the function $P (t) = 120, 000 - 20, 000 t$ over the interval $[0, 6]$ .

Step 2 ：Next, we apply the formula for continuous compounding to find the final value of the account after 6 years, using the rate of $4.67 %$ or $0.0467$ as a decimal.

Step 3 ：The final value of the account after 6 years, with continuous compounding, is calculated to be $476422.0073151354$ .

Step 4 ：We round the final value to the nearest dollar to get $476422$ .

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