Math Problem Statement

Use the continuous compound interest formula to find the indicated value. A = $71,000; P = $47,143; r = 9.1%; t = ?

Solution

The problem asks to use the continuous compound interest formula to find the time tt when given the final amount AA, principal PP, and interest rate rr.

The formula for continuous compound interest is:

A=PertA = P e^{rt}

where:

  • AA is the final amount,
  • PP is the initial principal,
  • rr is the interest rate (as a decimal),
  • tt is the time in years,
  • ee is the base of the natural logarithm (approximately 2.71828).

We are given:

  • A=71,000A = 71,000,
  • P=47,143P = 47,143,
  • r=9.1%=0.091r = 9.1\% = 0.091.

Now, we need to solve for tt. Follow these steps:

Step 1: Plug in the values

71,000=47,143e0.091t71,000 = 47,143 \cdot e^{0.091t}

Step 2: Divide both sides by 47,14347,143

71,00047,143=e0.091t\frac{71,000}{47,143} = e^{0.091t} 1.5067e0.091t1.5067 \approx e^{0.091t}

Step 3: Take the natural logarithm of both sides

ln(1.5067)=0.091t\ln(1.5067) = 0.091t 0.41070.091t0.4107 \approx 0.091t

Step 4: Solve for tt

t=0.41070.0914.51yearst = \frac{0.4107}{0.091} \approx 4.51 \, \text{years}

Thus, t4.51t \approx 4.51 years.

Would you like more details on any of the steps?

Here are 5 related questions to explore:

  1. How does continuous compound interest differ from regular compound interest?
  2. Can you solve for rr if tt is known?
  3. How would the formula change if interest were compounded monthly instead of continuously?
  4. What is the effect of doubling the principal PP on the time tt?
  5. How does the value of ee influence the behavior of the equation?

Tip: Always remember to convert percentage rates to decimals (e.g., 9.1% = 0.091) before using them in formulas.

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Math Problem Analysis

Mathematical Concepts

Exponential Functions
Continuous Compound Interest
Natural Logarithms

Formulas

Continuous Compound Interest Formula: A = P * e^(rt)
Natural Logarithm: ln(x)

Theorems

Properties of Exponential Functions
Logarithmic Inversion for Exponential Equations

Suitable Grade Level

Grades 10-12