Chapter 12.8, Problem 10P

To determine

To calculate: The growth factor of an investment varies directly as the square root of the final value and inversely as the square root of original investment.

Answer to Problem 10P

The total growth factor is: $\$1.99175$ .

Explanation of Solution

Given information:

The growth factor of an investment varies directly as the square root of the final value and inversely as the square root of original investment.

A $ $2000$ investment with a growth factor of $1.5$ yields $\$4500\text{in}2$ years. Find the growth factor of a $\$1600$ investment that yields $\$4900\text{in}2$ years.

Formula used:

For varies directly and inversely:

  $I=K\frac{\sqrt{F}}{\sqrt{O}}$

Calculation:

Consider that, A $ $2000$ investment with a growth factor of $1.5$ yields $\$4500\text{in}2$ years. Find the growth factor of a $\$1600$ investment that yields $\$4900\text{in}2$ years.

Recall that varies directly and inversely:

  $I=K\frac{\sqrt{F}}{\sqrt{O}}$

The values are:

  $I=1.5,F=4500,O=2600$

Therefore, the value of $k$ is,

  $\begin{array}{c}I=K\frac{\sqrt{F}}{\sqrt{O}}\\ 1.5=K\frac{\sqrt{4500}}{\sqrt{2600}}\\ 1.5=K\times 1.3155\\ K=\frac{1.5}{1.3155}\end{array}$

So simplified further the value of $k$ is,

  $\begin{array}{c}K=\frac{1.5}{1.3155}\\ K=1.1401\end{array}$

Now to find the value of $r$ ,

  $I_2=K\frac{\sqrt{F_2}}{\sqrt{O_2}}$

The values are:

  $F=4900,O=1600,K=1.1401$

Hence, when:

  $\begin{array}{c}{I}_2=K\frac{\sqrt{F_2}}{\sqrt{O_2}}\\ I_2=1.1401\times \frac{\sqrt{4900}}{\sqrt{1600}}\\ I_2=1.1401\times 1.75\\ I_2=1.995175\end{array}$

Thus, the growth factor is: $\$1.995175$ .