Find the information you're looking for at Westonci.ca, the trusted Q&A platform with a community of knowledgeable experts. Our platform provides a seamless experience for finding precise answers from a network of experienced professionals. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
To determine the length of time the money was invested, we need to use the compound interest formula. Given that the interest is compounded annually, the formula we will use is:
[tex]\[ A = P(1 + r)^t \][/tex]
where:
- \( A \) is the final amount of money (the balance after interest is applied),
- \( P \) is the principal amount (the initial amount of money),
- \( r \) is the annual interest rate (as a decimal),
- \( t \) is the number of years the money is invested.
We are given:
- \( P = 288 \) (the initial principal),
- \( r = 0.07 \) (the annual interest rate as a decimal),
- \( A = 581.23 \) (the final balance).
We need to solve for \( t \).
First, we start by rearranging the compound interest formula to solve for \( t \):
[tex]\[ \frac{A}{P} = (1 + r)^t \][/tex]
Next, we take the natural logarithm (ln) of both sides to get:
[tex]\[ \ln\left(\frac{A}{P}\right) = t \cdot \ln(1 + r) \][/tex]
Now, solve for \( t \):
[tex]\[ t = \frac{\ln\left(\frac{A}{P}\right)}{\ln(1 + r)} \][/tex]
Let's substitute the given values into the formula:
[tex]\[ t = \frac{\ln\left(\frac{581.23}{288}\right)}{\ln(1 + 0.07)} \][/tex]
Calculating the values inside the logarithm:
[tex]\[ \frac{581.23}{288} \approx 2.018826 \][/tex]
Now, plug this into the formula:
[tex]\[ t = \frac{\ln(2.018826)}{\ln(1.07)} \][/tex]
To find the natural logarithm values:
[tex]\[ \ln(2.018826) \approx 0.700353 \][/tex]
[tex]\[ \ln(1.07) \approx 0.067658 \][/tex]
Finally, divide these values to find \( t \):
[tex]\[ t \approx \frac{0.700353}{0.067658} \approx 10.378 \][/tex]
So, the money was invested for approximately 10.38 years.
[tex]\[ A = P(1 + r)^t \][/tex]
where:
- \( A \) is the final amount of money (the balance after interest is applied),
- \( P \) is the principal amount (the initial amount of money),
- \( r \) is the annual interest rate (as a decimal),
- \( t \) is the number of years the money is invested.
We are given:
- \( P = 288 \) (the initial principal),
- \( r = 0.07 \) (the annual interest rate as a decimal),
- \( A = 581.23 \) (the final balance).
We need to solve for \( t \).
First, we start by rearranging the compound interest formula to solve for \( t \):
[tex]\[ \frac{A}{P} = (1 + r)^t \][/tex]
Next, we take the natural logarithm (ln) of both sides to get:
[tex]\[ \ln\left(\frac{A}{P}\right) = t \cdot \ln(1 + r) \][/tex]
Now, solve for \( t \):
[tex]\[ t = \frac{\ln\left(\frac{A}{P}\right)}{\ln(1 + r)} \][/tex]
Let's substitute the given values into the formula:
[tex]\[ t = \frac{\ln\left(\frac{581.23}{288}\right)}{\ln(1 + 0.07)} \][/tex]
Calculating the values inside the logarithm:
[tex]\[ \frac{581.23}{288} \approx 2.018826 \][/tex]
Now, plug this into the formula:
[tex]\[ t = \frac{\ln(2.018826)}{\ln(1.07)} \][/tex]
To find the natural logarithm values:
[tex]\[ \ln(2.018826) \approx 0.700353 \][/tex]
[tex]\[ \ln(1.07) \approx 0.067658 \][/tex]
Finally, divide these values to find \( t \):
[tex]\[ t \approx \frac{0.700353}{0.067658} \approx 10.378 \][/tex]
So, the money was invested for approximately 10.38 years.
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.