Question

The number of exercises on Khan academy has increased rapidly since it began in 2006. The relationship between the elapsed time, t, in years, since Khan academy began, and the total number of its exercises, Eyear(t), is modeled by the following function: E year(t) = 100 ⋅ (1.7)ᵗ . Complete the following sentence about the monthly rate of change in the number of exercises. Round your answer to two decimal places.

A. The monthly rate of change is 0.06.
B. The monthly rate of change is 0.14.
C. The monthly rate of change is 1.70.
D. The monthly rate of change is 17.00.

Answer 1

The monthly rate of change in the number of exercises can be determined by finding the derivative of the given function and converting the time unit from years to months. The correct answer is B. The monthly rate of change is 0.14.

Step-by-step explanation:

The monthly rate of change in the number of exercises can be determined by finding the derivative of the function E year(t) = 100 ⋅ (1.7)ᵗ with respect to time, t. The derivative of the function is given by:

dE year(t)/dt = 100 ⋅ ln(1.7) ⋅ (1.7)ᵗ

To find the monthly rate of change, we need to convert the time unit from years to months. Since 1 year has 12 months, we divide the derivative by 12 to get the monthly rate of change:

Monthly rate of change = (100 ⋅ ln(1.7) ⋅ (1.7)ᵗ) / 12

Calculating this expression with t = 1, we find that the monthly rate of change is approximately 0.14. Therefore, the correct answer is B. The monthly rate of change is 0.14.