Assume you invest $250 at the end of each year for 10 years at...

Question

Assume you invest $250 at the end of each year for 10 years at an annual interest rate of $r$ . The amount of money in your account after 10 years is given by $A (r) = \frac{250 ({(1 + r)}^{10} - 1)}{r}$ . Assume your goal is to have $3500 in your account after 10 years.

b. Use a calculator to estimate the interest rate required to reach your financial goal.

b. Use a calculator to estimate the interest rate required to reach your financial goal.

Accepted Answer

Welcome back, everyone. You plan to invest $500 yearly into a college fund for your child for the next 12 years. The amount accumulated in the fund after 12 years can be calculated using the formula A of R equals 500 multiplied by 1 plus R raised to the power of 12 minus 1 divided by R, where R is the annual interest rate. Assume your goal is to have $9500 in the fund after 12 years. Use a calculator to determine the interest rate needed to achieve this goal. And were given for answer choices A 7.11%, B 6.13%, C 9.70%, and D 8.02%. So now let's rewrite the given formula. It says that the amount. A relative to R is equal to 500. Notice that this number comes from. The investment plan, it says you plan to invest $500 yearly, right? So this is why we use $500. Now, later on in the formula, we don't know what R is, and this is exactly what we're looking for. We want to identify. which is the annual interest rate. So we're going to leave R as a variable, so we get 1 plus R raised to the power of 12, right? We're using 12 because, well, we're given 12 years. It's already part of the formula. And now in the denominator of the fraction we once again have the same variable r. Now, we also know that a. is equal to $9500. This is given to us in the problem. It says that our goal is to have $9500 in the fund after 12 years. So we can set up our equation and solve for R using a calculator. The reason why we're using a calculator because, well, we're going to. Solve a complex equation and it is much easier to solve it using a calculator. So let's substitute a equals 9500 and let's set it equal to the right hand side of the equation which is. 500 multiplied by 1 plus R raise the power of 12 minus 1 divided by R. And this is the equation that we want to solve. If our calculator displays x instead of R, we can just set R equals X, and then our equation in the calculator would have a form of 9500 equals 500. Multiplied by 1 + X raise to the power of 12 minus 1 divided by X. Then we have to hit shift sole function and solve for X, and our calculator is going to give us a result x approximately equal to 0.080203126. Essentially, if we look at the answer choices, we want to get a percent expression. So we can say that rate R is going to be approximately equal to that number multiplied by 100, right, because this is the decimal form of the rate, and we're going to convert that interest rate into the percentage by multiplying the decimal form by 100, and this gives us 8.02, right, to decimal places. Therefore, the final answer to this problem corresponds to the answer choice D. Thank you for watching.

Key Concepts

Future Value of an Annuity

Interest Rate

Numerical Methods

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