Chapter 3.3, Problem 58E

A museum has decided to sell one of its paintings and to invest the proceeds. If the picture is sold between the years 2015 and 2025 and the money from the sale is invested in a bank account earning 2% interest per year compounded annually, then B(t), the balance in the year 2025, depends on the year, t, in which the painting is sold and the sale price P(t). If t is measured from the year 2015 so that 0 ≤ t ≤ 10 then

$B(t)=P(t){(1.02)}^{10-t}$

1. (a) Explain why B(t) is given by this formula.
2. (b) Show that the formula for B(t) is equivalent to

$B(t)={(1.02)}^{10}\frac{P(t)}{{(1.02)}^t}$.

1. (c) Find B′(5), given that P(5) = 150,000 dollars and P′(5) = 2000 dollars∕year.