To find the value of $L$ that maximizes Average Product of Labor (APL), we need to first define APL and then find its maximum. The APL is defined as:

$APL = \frac{Q}{L}$

Given $Q = 30L^2 - 0.5L^3$, we can substitute this into the formula for APL:

$APL = \frac{30L^2 - 0.5L^3}{L}$

Simplifying the expression:

$APL = 30L - 0.5L^2$

Next, to find the maximum APL, we need to take the derivative of APL with respect to $L$ and set it to zero:

$\frac{d(APL)}{dL} = 30 - L$

Setting the derivative equal to zero to find the critical point:

$30 - L = 0$

$L = 30$

To ensure this value of $L = 30$ is a maximum, we can check the second derivative of APL:

$\frac{d^2(APL)}{dL^2} = -1$

Since the second derivative is negative, $L = 30$ is indeed a maximum point for APL.

Therefore, the value of $L$ that maximizes APL is $L = 30$.

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Related Questions

1. How do you find the total product (TP) at the point where APL is maximized?
2. What is the value of APL when $L = 30$?
3. How do you find the marginal product of labor (MPL) from the given production function?
4. What is the economic interpretation of maximizing APL?
5. How does APL change as $L$ increases beyond 30?

Tip: The relationship between APL and Marginal Product of Labor (MPL) is crucial in understanding the efficiency of labor inputs in production.