Question

George invests a total of $5000 into various funds on the day his daughter is born. George’s investment balance can be estimated by the function f(t)=5000log100​(t+107)−75, where t is time in years since the investment was made.

Part A: During which time period did the account have the largest average rate of change? Assume t=0 on the day his daughter is born.
Part B: Interpret your result in Part A. Did George make good investments?
(A) [0,5], George made excellent investment decisions. The investment balance grew more and more rapidly over time.
(B) [10,15], George made moderately successful investment decisions. The investment balance grew over time, but at an increasingly slow rate.
(C) [15,20], George made disappointing investment decisions. He did not make any money because the investment balance remained the same over each time period.
(D) [5,10], George made bad investment decisions. He lost money over time.

Answer 1

The account had the largest average rate of change during the time period [0, -107]. George made excellent investment decisions as the investment balance grew more and more rapidly over time.

Step-by-step explanation:

Part A: To find the time period when the account had the largest average rate of change, we need to find the maximum value of the derivative of the function f(t). Taking the derivative of f(t), we get f'(t) = 5000 * (1/(t+107)). To find the maximum value, we need to solve the equation f'(t) = 0. Solving for t, we get t = -107. Since t represents time, we can ignore the negative value. Therefore, the account had the largest average rate of change during the time period [0, -107].

Part B: The interpretation of the result in Part A is that George made excellent investment decisions. The investment balance grew more and more rapidly over time, as indicated by the largest average rate of change during the time period [0, -107].