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 \( \boxed{476422} \).
Step 1 ：First, we need to calculate the total amount of money Gary will receive over the 6 years by integrating the function \( P(t) = 120,000 - 20,000t \) 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 \( \boxed{476422} \).
