Q=30L^2-0,5L^3 how do you find the value of L that maximises APL

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 \( L = 30 \).

Would you like more details or have any questions?

### 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.

Q=30L^2-0,5L^3 how do you find the value of L that maximises APL?

Learn how to find the value of L that maximizes the Average Product of Labor (APL) for the production function Q = 30L^2 - 0.5L^3. This problem involves calculus concepts such as optimization, derivatives, and applying the first and second derivative tests. Suitable for undergraduate economics or calculus students.

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