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Chapter 1: Problem 55

Find a mathematical model that represents the statement. (Determine the constant of proportionality.) \(A\) varies directly as \(r^{2} .(A=9 \pi \text { when } r=3 .)\)

Short Answer

Expert verified
The constant of proportionality is \(k = \pi\).

Step by step solution

01

Understand the Direct Variation Formula

Given a statement of direct variation like 'A varies directly as \(r^2\)', the formula can be written as \(A = kr^2\), where k is the constant of variation or proportionality.
02

Substitute the given values

The problem provides a specific instance where \(A = 9\pi\) when \(r = 3\). Substitute these values into the equation from step 1, to get \(9\pi = k * (3)^2\).
03

Solve for the constant of proportionality \(k\)

To solve for \(k\), divide both sides of the equation by \((3)^2\), which simplifies to \(k = \frac {9\pi} {(3)^2}\). Calculate the right side of the equation, to find the precise value of \(k\).

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Most popular questions from this chapter

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