Question

Certain pieces of antique furniture increased very rapidly in price in the 1970s and 1980s. For example, the value of a particular rocking chair is well approximated by V=150(1.65)^t where V is in dollars and t is the number of years since 1975. Find the rate, in dollars per year, at which the price is increasing.

Answer 1

The rate of change as a percentage is 65%.

The rate of change of the price is the derivative of the price function. To find the rate of change in dollars per year, we need to take the derivative of V with respect to t.

V'(t) = 150(1.65)^t * ln(1.65)

At t = 0, the rate of change is

V'(0) = 150(1.65)^0 * ln(1.65) = 150 * ln(1.65)

Therefore, the rate of change in dollars per year is 150 * ln(1.65).

To express this as a percentage, we need to divide by the current price of the rocking chair. The current price of the rocking chair is V(40) = 150(1.65)^40 = 10,508.11.

Therefore, the rate of change as a percentage is (150 * ln(1.65)) / 10,508.11 * 100% = 65.07%.

So the answer is 65%.