Chapter 1.10, Problem 54E

To determine

To Find: A mathematical model for the verbal statement and find constant of proportionality.

Answer to Problem 54E

The mathematical model of verbal statement is $A=\pi r^2$ and proportionality constant is $k=\pi$

Explanation of Solution

Given information:

‘A’ is directly proportional to $r^2$ and $\text{A=9}\pi \text{ when r}=3$

Concept and Formula Used:

Direct Variation: It is the relationship between two variable where one variable changes according to other directly at constant rate of vice versa.

Calculation:

The given statement is ‘A’ is directly proportional to $r^2$

The statement is mathematically represented as

  $A\propto r^2$

Let ‘k’ be the proportionality constant

Then

  $A=k{r}^2$

Now when

  $\begin{array}{l}\text{A=9}\pi \text{ when r}=3\\ 9\pi =k{\left(3\right)}^2\\ k=\frac{9\pi }{9}\\ k=\pi \end{array}$

Therefore

  $A=\pi r^2$

Conclusion:

The mathematical model of verbal statement is $A=\pi r^2$ and proportionality constant is $k=\pi$