Suppose that $y$ is proportional to the $4^{\text {th }}$ power of $x$, and that $y=13$ when $x=2$. What is $y$ when $x=5$ ? Round your answer to two decimal places if necessary.
Answer

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Step 1 ：The problem states that \(y\) is proportional to the \(4^{\text{th}}\) power of \(x\). This means that \(y = kx^4\) for some constant \(k\).

Step 2 ：We can find the value of \(k\) using the given condition that \(y=13\) when \(x=2\).

Step 3 ：Substituting \(x=2\) and \(y=13\) into the equation \(y = kx^4\), we get \(13 = k(2^4)\). Solving for \(k\), we find that \(k = 0.8125\).

Step 4 ：Now that we have the value of \(k\), we can substitute \(x=5\) into the equation to find the corresponding value of \(y\).

Step 5 ：Substituting \(x=5\) and \(k=0.8125\) into the equation \(y = kx^4\), we get \(y = 0.8125(5^4)\).

Step 6 ：Solving for \(y\), we find that \(y = 507.8125\).

Step 7 ：Final Answer: The value of \(y\) when \(x=5\) is \(\boxed{507.81}\) when rounded to two decimal places.