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Chapter 3: Problem 14

(a) solve for \(P\) and (b) solve for \(t\). $$A=P\left(1+\frac{r}{n}\right)^{n t}$$

Short Answer

Expert verified
The short answers are \( P = \frac{A}{(1+\frac{r}{n})^{nt}} \) and \( t = \frac{ln(\frac{A}{P})}{n \cdot ln(1+\frac{r}{n})} \)

Step by step solution

01

Isolate \(P\)

To isolate \(P\), we begin by dividing both sides by \((1+\frac{r}{n})^{nt}\), this gives us: \[ P = \frac{A}{(1+\frac{r}{n})^{nt}} \]
02

Solve for \(t\)

Solving for \(t\) is a bit more complicated. We'll start by dividing both sides by \(P\), which gives us: \[ (1+\frac{r}{n})^{nt} = \frac{A}{P} \] Then we take natural logarithm (ln) on both sides of the equation, which helps us to bring \(t\) down from the exponent: \[ nt \cdot ln(1+\frac{r}{n}) = ln(\frac{A}{P}) \] Now, isolate \(t\) by dividing both sides by \(n \cdot ln(1+\frac{r}{n})\): \[ t = \frac{ln(\frac{A}{P})}{n \cdot ln(1+\frac{r}{n})} \]. Remember that ln(x) is the natural logarithm and that \(ln(ab) = ln(a) + ln(b)\) and \(ln(a^b) = b \cdot ln(a)\)

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Most popular questions from this chapter

A logarithmic model has the form ________ or ________.

A logistic curve is also called a ________ curve.

The number \(y\) of hits a new search-engine website receives each month can be modeled by \(y=4080 e^{k t},\) where \(t\) represents the number of months the website has been operating. In the website's third month, there were 10,000 hits. Find the value of \(k,\) and use this value to predict the number of hits the website will receive after 24 months.

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