Question

a varies directly as the square of b and inversely as the square of c. if a = 95 when b = 8 and c = 4, find a if b = 7 and c = 9. round your answer to two decimal places if necessary.
answer
a =

Explanation:

Step1: Write the variation equation

Since $a$ varies directly as the square of $b$ and inversely as the square of $c$, the equation is $a = k\frac{b^{2}}{c^{2}}$, where $k$ is the constant of variation.

Step2: Find the value of $k$

Substitute $a = 95$, $b = 8$ and $c = 4$ into the equation $a = k\frac{b^{2}}{c^{2}}$. We get $95=k\frac{8^{2}}{4^{2}}$. First, calculate $\frac{8^{2}}{4^{2}}=\frac{64}{16} = 4$. Then, $95 = 4k$. Solving for $k$, we have $k=\frac{95}{4}=23.75$.

Step3: Find $a$ for new $b$ and $c$

Now that $k = 23.75$, and we want to find $a$ when $b = 7$ and $c = 9$. Substitute these values into $a = k\frac{b^{2}}{c^{2}}$. So $a=23.75\times\frac{7^{2}}{9^{2}}$. Calculate $7^{2}=49$ and $9^{2} = 81$. Then $a=23.75\times\frac{49}{81}=\frac{23.75\times49}{81}=\frac{1163.75}{81}\approx14.37$.

Answer:

$14.37$