Chapter 3.4, Problem 14CFU

a.

To determine

To find: the formula for r.

Answer to Problem 14CFU

  $r=\sqrt[3]{\frac{B}{1000}}-1$

Explanation of Solution

Given information:

The function is as follows:

  $B\left(r\right)=1000{\left(1+r\right)}^3$

Here B is the balance in the account, r is the interest rate compounded annually, and the power denotes the number of years, and the amount deposited is $1000.

Calculation:

The formula for r can be evaluated as follows:

  $\begin{array}{c}B\left(r\right)=1000{\left(1+r\right)}^3\left[\text{Let }B\left(r\right)=B\right]\\ B=1000{\left(1+r\right)}^3\left[\text{Solve for }r\right]\\ \frac{B}{1000}={\left(1+r\right)}^3\\ \sqrt[3]{\frac{B}{1000}}=1+r\\ \sqrt[3]{\frac{B}{1000}}-1=r\\ or\\ r=\sqrt[3]{\frac{B}{1000}}-1\end{array}$

Thus the formula is $r=\sqrt[3]{\frac{B}{1000}}-1$ .

b.

To determine

To find: r (in decimals) for the given condition.

Answer to Problem 14CFU

  $f^{-1}\left(x\right)=2x+10$

Explanation of Solution

Given information:

  $\begin{array}{l}B=\$1100\\ t=3\text{years}\end{array}$

The formula evaluated for r compounded for 3 years is shown below.

  $r=\sqrt[3]{\frac{B}{1000}}-1$

Calculation:

Put the value of B in the given formula and solve r as follows:

  $\begin{array}{c}r=\sqrt[3]{\frac{1100}{1000}}-1\\ =1.0323-1\\ =0.0323\end{array}$

Thus the interest rate is 0.0323.