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Sagot :
To solve the problem of finding the interest rate Adam earned, we will follow a step-by-step approach using the given exponential growth formula [tex]\( A = P e^{rt} \)[/tex], where:
- [tex]\(A\)[/tex] is the ending amount
- [tex]\(P\)[/tex] is the principal (initial amount)
- [tex]\(r\)[/tex] is the interest rate
- [tex]\(t\)[/tex] is time
Given:
- [tex]\( A = 7649 \)[/tex]
- [tex]\( P = 100 \)[/tex]
- [tex]\( t = 10 \)[/tex] years
We need to find [tex]\( r \)[/tex], the interest rate.
### Step-by-Step Solution:
1. Set up the equation with the given values:
[tex]\[ 7649 = 100 \cdot e^{r \cdot 10} \][/tex]
2. Isolate the exponential term:
Divide both sides of the equation by 100 to simplify:
[tex]\[ \frac{7649}{100} = e^{10r} \][/tex]
[tex]\[ 76.49 = e^{10r} \][/tex]
3. Solve for [tex]\( r \)[/tex]:
To solve for [tex]\( r \)[/tex], we need to apply the natural logarithm (ln) to both sides of the equation. The natural logarithm is the inverse function of the exponential function [tex]\( e \)[/tex]:
[tex]\[ \ln(76.49) = \ln(e^{10r}) \][/tex]
4. Simplify the logarithmic expression:
Using the property of logarithms [tex]\( \ln(e^x) = x \)[/tex], we simplify the right-hand side:
[tex]\[ \ln(76.49) = 10r \][/tex]
5. Isolate [tex]\( r \)[/tex]:
[tex]\[ r = \frac{\ln(76.49)}{10} \][/tex]
6. Calculate [tex]\( r \)[/tex]:
We can approximate the value:
[tex]\[ \ln(76.49) \approx 4.336 \][/tex]
[tex]\[ r = \frac{4.336}{10} \approx 0.4336 \][/tex]
7. Convert [tex]\( r \)[/tex] to a percentage:
Convert the decimal form of the interest rate to a percentage by multiplying by 100:
[tex]\[ r \times 100 \approx 0.4336 \times 100 \approx 43.36\% \][/tex]
### Simplifying further, we can represent the solution with the correct answer:
Given the multiple choices:
- A. [tex]\(3\%\)[/tex]
- B. [tex]\(20\%\)[/tex]
- C. [tex]\(28\%\)[/tex]
- D. [tex]\(5\%\)[/tex]
Our calculated interest rate [tex]\(43.36\%\)[/tex] does not match any of the given choices due to potential typographical error in the provided options.
So, the correct interpretation of the interest rate based on the provided calculation is approximately:
[tex]\[ r \approx 43.36\% \][/tex]
Thus, the interest rate Adam earned over 10 years is approximately [tex]\(43.36\%\)[/tex]. Despite the mismatch, this is the step-by-step solution.
- [tex]\(A\)[/tex] is the ending amount
- [tex]\(P\)[/tex] is the principal (initial amount)
- [tex]\(r\)[/tex] is the interest rate
- [tex]\(t\)[/tex] is time
Given:
- [tex]\( A = 7649 \)[/tex]
- [tex]\( P = 100 \)[/tex]
- [tex]\( t = 10 \)[/tex] years
We need to find [tex]\( r \)[/tex], the interest rate.
### Step-by-Step Solution:
1. Set up the equation with the given values:
[tex]\[ 7649 = 100 \cdot e^{r \cdot 10} \][/tex]
2. Isolate the exponential term:
Divide both sides of the equation by 100 to simplify:
[tex]\[ \frac{7649}{100} = e^{10r} \][/tex]
[tex]\[ 76.49 = e^{10r} \][/tex]
3. Solve for [tex]\( r \)[/tex]:
To solve for [tex]\( r \)[/tex], we need to apply the natural logarithm (ln) to both sides of the equation. The natural logarithm is the inverse function of the exponential function [tex]\( e \)[/tex]:
[tex]\[ \ln(76.49) = \ln(e^{10r}) \][/tex]
4. Simplify the logarithmic expression:
Using the property of logarithms [tex]\( \ln(e^x) = x \)[/tex], we simplify the right-hand side:
[tex]\[ \ln(76.49) = 10r \][/tex]
5. Isolate [tex]\( r \)[/tex]:
[tex]\[ r = \frac{\ln(76.49)}{10} \][/tex]
6. Calculate [tex]\( r \)[/tex]:
We can approximate the value:
[tex]\[ \ln(76.49) \approx 4.336 \][/tex]
[tex]\[ r = \frac{4.336}{10} \approx 0.4336 \][/tex]
7. Convert [tex]\( r \)[/tex] to a percentage:
Convert the decimal form of the interest rate to a percentage by multiplying by 100:
[tex]\[ r \times 100 \approx 0.4336 \times 100 \approx 43.36\% \][/tex]
### Simplifying further, we can represent the solution with the correct answer:
Given the multiple choices:
- A. [tex]\(3\%\)[/tex]
- B. [tex]\(20\%\)[/tex]
- C. [tex]\(28\%\)[/tex]
- D. [tex]\(5\%\)[/tex]
Our calculated interest rate [tex]\(43.36\%\)[/tex] does not match any of the given choices due to potential typographical error in the provided options.
So, the correct interpretation of the interest rate based on the provided calculation is approximately:
[tex]\[ r \approx 43.36\% \][/tex]
Thus, the interest rate Adam earned over 10 years is approximately [tex]\(43.36\%\)[/tex]. Despite the mismatch, this is the step-by-step solution.
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