Pergunta

A loan of 16,000 is made at 5% interest, compounded annually.After how many years will the amount due reach 28,000 or more? (Use the calculator provided if necessary.) Write the smallest possible whole number answer. square square
Solução

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JegorElite · Tutor por 8 anos
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The amount due will reach $28,000 or more after 15 years.
Explicação
## Step 1<br />The problem involves compound interest, which can be calculated using the formula:<br />### \(A = P(1 + \frac{r}{n})^{nt}\)<br />where:<br />- \(A\) is the amount of money accumulated after \(n\) years, including interest.<br />- \(P\) is the principal amount (the initial amount of money).<br />- \(r\) is the annual interest rate (in decimal).<br />- \(n\) is the number of times that interest is compounded per year.<br />- \(t\) is the time the money is invested for, in years.<br /><br />## Step 2<br />In this problem, the principal amount \(P\) is $16,000, the annual interest rate \(r\) is 5% or 0.05, and the interest is compounded annually, so \(n\) is 1. We are asked to find the time \(t\) when the amount \(A\) will reach $28,000 or more.<br /><br />## Step 3<br />We can rearrange the formula to solve for \(t\):<br />### \(t = \frac{\log(\frac{A}{P})}{n \cdot \log(1 + \frac{r}{n})}\)<br /><br />## Step 4<br />Substitute the given values into the formula:<br />### \(t = \frac{\log(\frac{28000}{16000})}{1 \cdot \log(1 + \frac{0.05}{1})}\)<br /><br />## Step 5<br />Calculate the value of \(t\). The result is approximately 14.2067 years.<br /><br />## Step 6<br />Since the problem asks for the smallest possible whole number, we round up to the next whole number, which is 15.
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