Question

The average profit a local store owner earns on a given day is $830 and is growing exponentially at a rate of 58% per year. Write a function to represent profit after t years, where the monthly rate of change can be found from a constant in the function. Round all coefficients in the function to four decimal places. Also, determine the percentage rate of change per month, to the nearest hundredth of a percent.

Answer 1

The profit function after t years with an initial profit of $830 and a 58% annual growth rate is P(t) = 830 * e(0.58/12)t, and the monthly percentage rate change is approximately 4.83%.

Step-by-step explanation:

The function to represent profit after t years given an initial average profit of $830 and an exponential growth rate of 58% per year can be written in the general form of the exponential function P(t) = P0 * ert where P0 is the initial amount, r is the rate of growth, and t is the time in years. Since we are looking for a monthly rate, we'll first convert the annual rate to a monthly rate. To find the monthly growth rate, divide the annual rate by 12. The formula becomes P(t) = 830 * e(0.58/12)t. Now, to calculate the percentage rate of change per month, we can use the formula for continuous growth, which is the monthly rate times 100. The monthly rate is 0.58 divided by 12, yielding approximately 0.0483. Therefore, the percentage rate change per month is 0.0483 * 100 = 4.83%.