Final answer:
To find the time it takes for an investment to reach $7,500 with 10% interest compounded continuously, use T = 10 · ln(1.5). The inverse function reflects the growth factor after a given time, with f^{-1}(8) indicating the growth factor after the time corresponding to a value of 8.
Step-by-step explanation:
To find how long it will take for the balance in the account to reach $7,500, we will use the formula provided, where T(time) is given by T = k · ln(A), with k as a constant related to the interest rate, and A as the factor by which the initial amount will grow. Since the interest rate is 10% and is compounded continuously, and we want the account to grow by a factor of $7,500/$5,000 = 1.5, we can solve for T using the formula T = 10 · ln(1.5).
To determine how long it takes for the investment to increase to any given amount, we can find the inverse function. If f(x) = k · ln(x), then the inverse function f^{-1}(y) will solve for x when given a value for y. For f^{-1}(8) = e^{8/10}, this represents the factor by which the initial amount has grown after a time period corresponding to 8 (in the same time units used for the constant k). Specifically, f^{-1}(8) tells us by how much the original amount has increased after the amount of time it takes for the investment to grow by a value of 8 on the original scale.