# An investment grows according to the exponential equation y = 15,000(1.07)^x, where x is the...

## Question:

An investment grows according to the exponential equation {eq}y = 15,000(1.07)^x {/eq}, where x is the number of years invested. Which of the following statements is true?

A. The investment will continue to grow at a rate of {eq}7\% {/eq} per year compounded quarterly.

B. The investment will increase by {eq}\\$1050 {/eq} per year.

C. The investment will more than double within 12 years.

D. The investment will triple within 15 years.

## Exponential Functions: Investment Calculations

Assume that we wish to calculate the investment after a specific time (in years) {eq}P_t {/eq} with an initial investment {eq}P_o {/eq} that follows an exponential model.

This model can be written as {eq}P_t = P_o(1 + \frac{p}{100})^t {/eq}, where {eq}p {/eq} is the percentage interest earned.

Given:

{eq}y = 15,000(1.07)^x {/eq}

a. Let's check our initial statement. This statement is false, since our investment grows at a rate of 7 % compounded yearly, which is indicated by the number of years, {eq}x {/eq}.

b. The statement (b) implies that the increase is steady, as though we were working with a linear function. However, this is not true. An exponential growth follows an increase in growth proportionally every year.

c. Let's identify when our investment will double. We calculate this by:

{eq}\dfrac{y}{15,000} = 2 = (1.07)^x {/eq}

Solving for the number of years:

{eq}x = \dfrac{\ln(2)}{\ln(1.07)}\approx 10 {/eq}

Therefore, the investment will more than double within 12 years.

d. Let's check when the investment will triple:

{eq}\dfrac{y}{15,000} = 3 = (1.07)^x\\ x = \dfrac{\ln(3)}{\ln(1.07)}\approx 16 {/eq}

Thus, the investment will triple in about 16 years.

We can conclude that the statement {eq}\boxed{C} {/eq} is true. 