A museum has decided to sell one of its paintings and to

Question

Question: A museum has decided to sell one of its paintings and to invest the proceeds. If the picture is sold between the years 2000 and 2020 and the money from the sale is invested in a bank account earning 5% interest per year compounded annually, then B(t), the balance in the year 2020, depends on the year, t, in which the painting is sold and the sale price P(t). If t is measured from the year 2000 so that 0

B(t) = P(t)(1.05)^{20 - t}

(a) Explain why B(t) is given by this formula.

(b) Show that the formula for B(t) is equivalent to

B(t) = (1.05)²⁰ (P(t)/(1.05)^t)

(c) Find B'(10), given that P(10) = 150,000 and P'(10) = 5000.