Math Problem Statement

Use the compound interest formula Aequals​P(1plus​r)Superscript t and the given information to solve for r. Aequals​$2200​, Pequals​$1700​, tequals4 Question content area bottom Part 1 requals    enter your response here​% ​(Round to the nearest​ hundredth.)

Solution

The compound interest formula is:

A=P(1+r)tA = P(1 + r)^t

Where:

  • AA is the final amount (2200 in this case),
  • PP is the principal amount (1700 in this case),
  • rr is the annual interest rate (which we are solving for),
  • tt is the time the money is invested or borrowed for (4 years here).

We are given:

  • A=2200A = 2200
  • P=1700P = 1700
  • t=4t = 4

Now, plug the values into the formula:

2200=1700(1+r)42200 = 1700(1 + r)^4

Step 1: Divide both sides of the equation by 1700 to isolate (1+r)4(1 + r)^4:

22001700=(1+r)4\frac{2200}{1700} = (1 + r)^4

1.2941=(1+r)41.2941 = (1 + r)^4

Step 2: Take the fourth root of both sides to eliminate the exponent:

1.29414=1+r\sqrt[4]{1.2941} = 1 + r

1.0652=1+r1.0652 = 1 + r

Step 3: Subtract 1 from both sides to solve for rr:

r=1.06521r = 1.0652 - 1

r=0.0652r = 0.0652

Step 4: Convert this decimal to a percentage by multiplying by 100:

r=6.52%r = 6.52\%

So, the annual interest rate rr is approximately 6.52%.

Let me know if you'd like further details or have any questions!

Here are some related questions to explore further:

  1. How would the interest rate change if the time tt were different, say 5 years?
  2. How do you calculate compound interest if interest is compounded more frequently than annually?
  3. What would the final amount AA be if the principal was $2000 with the same rate and time?
  4. How do you derive the compound interest formula from the simple interest formula?
  5. How does compound interest differ from simple interest in terms of growth over time?

Tip: When solving for an unknown in an exponentiation equation like this, remember to isolate the base first and then apply the appropriate root or logarithmic function to solve.

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Math Problem Analysis

Mathematical Concepts

Compound Interest
Algebra
Exponents

Formulas

A = P(1 + r)^t

Theorems

Compound Interest Theorem

Suitable Grade Level

Grades 9-12