Math Problem Statement

Jerome will be buying a used car for $7,000 in 2 years. How much money should he ask his parents for now so that, if he invests it at 4% compounded continuously, he will have enough to buy the car? (Round to the nearest cent as needed.)

Solution

To solve this problem, we use the formula for continuous compounding:

A=PertA = P e^{rt}

Where:

  • AA is the future amount ($7,000 in this case),
  • PP is the present amount (the amount Jerome should ask for),
  • rr is the annual interest rate (4% = 0.04),
  • tt is the time in years (2 years),
  • ee is the mathematical constant approximately equal to 2.718.

We solve for PP:

P=AertP = \frac{A}{e^{rt}}

Substituting the values: P=7000e0.042P = \frac{7000}{e^{0.04 \cdot 2}}

Let me calculate this for you:

P=7000e0.0870001.0832876463.97P = \frac{7000}{e^{0.08}} \approx \frac{7000}{1.083287} \approx 6463.97

So, Jerome should ask for $6,463.97.

Let me know if you'd like a breakdown or have further questions!

5 Related Questions:

  1. How would the amount change if the interest rate were 5% instead of 4%?
  2. What happens to the amount needed if the time period changes to 3 years?
  3. How would the result differ if the interest were compounded annually instead of continuously?
  4. What formula would you use for simple interest instead of continuous compounding?
  5. What if Jerome wanted to save for a $10,000 car instead? How much would he need now?

Tip:

When working with continuous compounding, remember that small changes in the rate or time can have a significant effect due to exponential growth.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Continuous Compounding
Exponential Growth
Algebra

Formulas

A = P * e^(rt)
P = A / e^(rt)

Theorems

Exponential Growth Formula

Suitable Grade Level

Grades 11-12 (Precalculus)