Chapter 7.4, Problem 89PFA

a.

To determine

Find the interest rate of the amount.

Answer to Problem 89PFA

The rate of interest of the amount is $2\%$ .

Explanation of Solution

Given: It is given in the question that the initial investment is $\$5780$ and the value will be $\$6080$ after two years and the function for the rate will be $r=\sqrt[t]{\frac{A}{P}}-1$ .

Concept Used:

In this, use the concept of putting the value in the equation and get the answer.

Calculation: Here, $p=5750,A=6080,t=2$

  $\begin{array}{l}r=\sqrt[t]{\frac{A}{P}}-1\\ r=\sqrt[2]{\frac{6080}{5750}}-1\\ r=\sqrt[2]{1.05}-1\\ r=1.02-1\\ r=0.02\\ r=0.02\times 100\\ r=2\%\end{array}$

Conclusion:

The rate of interest is $2\%$ .

b.

To determine

Find the function for P.

Answer to Problem 89PFA

The function is $P=\frac{{(r+1)}^t}{A}$ .

Explanation of Solution

Given:

It is given in the question that the function is $r=\sqrt[t]{\frac{A}{P}}-1$ .

Concept Used:

In this, use the concept of solving the function and get P.

Calculation: The given function in terms of r is $r=\sqrt[t]{\frac{A}{P}}-1$ .

  $\begin{array}{l}r=\sqrt[t]{\frac{A}{P}}-1\\ r+1=\sqrt[t]{\frac{A}{P}}\\ $ \frac{A}{P}={(r+1)}^t$P=\frac{{(r+1)}^t}{A}\end{array}$

Conclusion:

The function is $P=\frac{{(r+1)}^t}{A}$ .

c.

To determine

Calculate the initial investment of the value.

Answer to Problem 89PFA

The initial investment will be $1316$ .

Explanation of Solution

Given: It is given in the question that the initial value is $\$4000$ and the time is $12$ years and the rate of interest is $3.4\%$ .

Concept Used:

In this, use the formula of P , i.e. $P=\frac{{(r+1)}^t}{A}$ .

Calculation: Here, the formula be $P=\frac{{(r+1)}^t}{A}$ .

  $\begin{array}{l}P=\frac{{(r+1)}^t}{A}\\ P=\frac{{(3.4+1)}^{12}}{4000}\\ P=\frac{{(4.4)}^{12}}{4000}\\ P=\frac{5265480}{4000}\\ P=1316\end{array}$

Conclusion:

The initial investment will be $1316$ .