Pre Calculus Question

Question

How long will it take for $5,000 to double in the following investments? round to the nearest tenth of a year.

Simple interest earning 5.9%APR: 

Compounded monthly earning 5.9%APR: 

compounded continuously earning 5.9%APR: 

How much money would be in the account after 10 years for each of the following investments. Round to the cent.



Simple interest earning 5.9%APR: $

.Compounded monthly earning 5.9%APR: $

.Compounded continuously earning 5.9%APR: $

Answer 1

You will want to use the formula A = P(1 + r)^t to find simple interest. In place of A put 10,000, in place of P put 5000, in place of r put 0.059 and in place of t put t since you are finding that value. After doing this, you have 10,000 = 5000(1 + .059)^t. Divide both sides by 5000 and you get 2 = 1.059^t. Then take the log of both sides: log 2 = log (1.059)^t. Use your rule of logs to simplify the right side to log 2 = t log 1.059. This simplifies to 0.30103 = 0.02490t. Divide both sides by 0.02490 and you get t = 12.1 years.

To work out compound interest, use the formula A = P(1 + r/n)^(nt). Substitute 10,000 in place of A, 5000 in place of P, 0.059 in place of r and 12 in place of n since it is compounded monthly and there are 12 months in a year. We then get 10,000 = 5000(1 + .059/12)^(12t). Divide both sides by 5000 and you get 2 = 1.00492^(12t). Then log both sides and you get log 2 = log 1.00492^(12t). Use your rule of logs to simplify to log 2 = 12t log 1.00492. Then divide both sides by log 1.00492. This gives log 2 / log 1.00492= 12t. Simplify the left side to get 141.23 = 12t. Divide by 12 and you get 11.8 years

This time use the formula A = Pe^rt since it is compounded continuously. Substitute 10,000 in place of A, 5000 in place of P, and 0,059 in place of r. This gives 10,000 = 5000 e^0.059t. Again divide both sides by 5000 and you get 2 = e^0.059t. Take the natural log of both sides to get ln 2 = ln e^0.059t . Simplifying the right side gives you ln 2 = 0.059t. Then divide both sides by 0.059 and you get 11.7 years.

To find the amount in the account after 10 years, use A = P(1 + r)^t. Substitute 5000 in place of P, 0059 in place of r and 10 in place of n. This gives A = 5000(1 + 0.059)^10. Multiply that out and you get 8870.12.

To find the amount using compound interest use A = P(1 + r/n)^nt. Substitute 5000 in place of P, 0.059 in place of r, 12 in place of n since it compounded monthly, and 10 in place of t since we want amount after 10 years. This gives us A = 5000(1 + 0.059/12)^(12*10). Using your calculator you get 9006.91 after 10 years.

To find the amount after compounding continuously, use the formula A = Pe^(rt). Substitute 5000 in place of P, 0.059 in place of r, and 10 in place of t since we want amount after 10 years. This gives us A = 5000 e^(0.059*10). Use your calculator and you will find A to be 9019.94.