The annual net sales (revenue) earned by Nintendo Co., Ltd., in the fiscal years ending March 31, 1995 to March 31, 2003, can be approximated by $R ( t ) = - 4 t ^ { 2 } + 10 t + 530$ billion yen per year $( - 6 \leq t \leq 3 )$ where t is time in years ( $t = 0$ represents March 31, 2000). Suppose that, from April 1994 on, Nintendo invested its revenue in an investment yielding 4% compounded continuously. What, to the nearest 100 billion, would the total value of Nintendo's revenue have been by the end of March 2003
A)Total revenue 500 billion yen
B) Total revenue 2,600 billion yen
C) Total revenue 1,000 billion yen
D) Total revenue 5,100 billion yen
E) Total revenue 51,200 billion yen

Question

The annual net sales (revenue) earned by Nintendo Co., Ltd., in the fiscal years ending March 31, 1995 to March 31, 2003, can be approximated by $R ( t ) = - 4 t ^ { 2 } + 10 t + 530$ billion yen per year $( - 6 \leq t \leq 3 )$ where t is time in years ( $t = 0$ represents March 31, 2000). Suppose that, from April 1994 on, Nintendo invested its revenue in an investment yielding 4% compounded continuously. What, to the nearest 100 billion, would the total value of Nintendo's revenue have been by the end of March 2003   
A)Total revenue   500 billion yen
B) Total revenue   2,600 billion yen
C) Total revenue   1,000 billion yen
D) Total revenue   5,100 billion yen
E) Total revenue   51,200 billion yen

Quizplus · Accepted Answer

To find the total value of Nintendo's revenue by the end of March 2003, we need to calculate the integral of the revenue function compounded continuously at a rate of 4%. The formula for continuous compounding is $A = Pe^{rt}$, where $A$ is the amount of money accumulated after n years, including interest, $P$ is the principal amount (initial investment), $r$ is the annual interest rate (decimal), and $t$ is the time the money is invested for in years.Given $R(t) = -4t^2 + 10t + 530$, we need to integrate this function from $t = -6$ to $t = 3$ (covering the years from the end of March 1995 to the end of March 2003) and apply the continuous compounding formula. The integral gives us the total revenue over the period, and then we apply the continuous compounding formula to find the total value.The integral of $R(t)$ over the period $t = -6$ to $t = 3$ gives us the total revenue without compounding. However, to find the total value with continuous compounding, we would integrate the function $R(t)e^{0.04t}$ over the same interval. This calculation is complex and typically requires numerical methods or a calculator, and the exact steps are not shown here due to the nature of the question.The correct answer, after performing the necessary calculations, indicates that the total revenue, compounded continuously at 4% from April 1994 to the end of March 2003, would be closest to the option given by D) Total revenue ≈ 5,100 billion yen. This takes into account both the revenue earned each year and the continuous compounding of this revenue at a rate of 4%.

The Annual Net Sales (Revenue) Earned by Nintendo Co R(t)=−4t2+10t+530R ( t ) = - 4 t ^ { 2 } + 10 t + 530R(t)=−4t2+10t+530

The Annual Net Sales (Revenue) Earned by Nintendo Co $R ( t ) = - 4 t ^ { 2 } + 10 t + 530$