A company finds that the cost, in dollars, of producing \( t \) handbags is given by \( C(t)=465+25 t+10 t^{2} \). Find the rate at which average cost, \( \frac{C(t)}{t} \), is changing when 200 handbags have been produced. (Round to the nearest cent.) The average cost is ? by handbags are produced.

To find the rate at which the average cost is changing, we first need to express the average cost function, \( A(t) = \frac{C(t)}{t} \). From the cost function \( C(t) = 465 + 25t + 10t^2 \), we can derive: \[ A(t) = \frac{465 + 25t + 10t^2}{t} = \frac{465}{t} + 25 + 10t \] Next, we differentiate \( A(t) \) with respect to \( t \) using the quotient rule. \[ A'(t) = -\frac{465}{t^2} + 10 \] Now, let's calculate \( A'(t) \) when \( t = 200 \): \[ A'(200) = -\frac{465}{200^2} + 10 = -\frac{465}{40000} + 10 = -0.011625 + 10 \approx 9.988375 \] Rounding it to the nearest cent gives us: The average cost is decreasing by approximately \( \$9.99 \) when 200 handbags are produced.