# Chapter 6 - Section 6.6 - Variation - 6.6 Exercises - Page 418: 37

$a=4$

#### Work Step by Step

Recall: If $y$ varies directly as the $n^{\text{th}}$ power of $x$, then the direct variation's equation is $y=kx^n$ where $k$ is the constant of variation. Since $a$ varies directly as the square of $b$, then the equation of the direct variation, with $k$ as the constant of variation, is: $$a=kb^2$$ When $a=4$, $b=3$. Substitute these into the equation above to obtain: \begin{align*} a&=kb^2\\\\ 4&=k(3^2)\\\\ 4&=9k\\\\ \frac{4}{9}&=k \end{align*} Thus, the equation fo the direct variation is: $$a=\frac{4}{9}b^2$$ To find the value of $a$ when $b=2$, substitute $2$ to $b$ in the equation above to obtain: \begin{align*} a&=\frac{4}{9}b^2\\\\ a&=\frac{4}{9} \cdot (3^2)\\\\ a&=\frac{4}{9} \cdot 9\\\\ a&=4 \end{align*}

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