Step 1 :Given the function \(f(t)=7 t^{6}\) at \(t=20\) and \(t=50\)
Step 2 :First, find the derivative of the function, \(f'(t)\)
Step 3 :\(f'(t) = 42t^{5}\)
Step 4 :Then, find the logarithmic derivative at the given points using the formula \(\frac{f'(t)}{f(t)}\)
Step 5 :At \(t=20\), the logarithmic derivative is \(\frac{3}{10}\)
Step 6 :At \(t=50\), the logarithmic derivative is \(\frac{3}{25}\)
Step 7 :Finally, find the percentage rate of change at the given points using the formula \(\frac{f'(t)}{f(t)} \times 100\%\)
Step 8 :At \(t=20\), the percentage rate of change is \(30\%\)
Step 9 :At \(t=50\), the percentage rate of change is \(12\%\)
Step 10 :Final Answer: The percentage rate of change of the function at \(t=20\) is \(\boxed{30\%}\) and at \(t=50\) is \(\boxed{12\%}\)