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.