To find the marginal-revenue product (MRP) for the given value of $m = 90$, we follow these steps:

1. Define the functions:

• The production function is given by: $q = \frac{200m - m^2}{20}$
• The price function is given by: $p = -0.3q + 80$

The marginal-revenue product (MRP) is the additional revenue generated from hiring one more worker. It can be calculated as: $\text{MRP} = \frac{dR}{dm}$ Where $R$ is the total revenue: $R = p \cdot q$

2. Find $q$ as a function of $m$:

The production function is already given as: $q(m) = \frac{200m - m^2}{20}$ We differentiate this to find $\frac{dq}{dm}$: $\frac{dq}{dm} = \frac{1}{20} (200 - 2m) = \frac{200 - 2m}{20}$

3. Express $p$ in terms of $m$:

Since $p = -0.3q + 80$, substitute the expression for $q$ into the price function: $p(m) = -0.3 \cdot \left(\frac{200m - m^2}{20}\right) + 80$ Simplifying:

= \frac{-60m + 0.3m^2}{20} + 80 = \frac{-3m + 0.015m^2}{1} + 80$$ So: $$p(m) = 80 - 3m + 0.015m^2$$ ### 4. Find the revenue $$R$$: The total revenue is: $$R(m) = p(m) \cdot q(m)$$ Substitute the expressions for $$p(m)$$ and $$q(m)$$: $$R(m) = \left(80 - 3m + 0.015m^2\right) \cdot \left(\frac{200m - m^2}{20}\right)$$ ### 5. Differentiate $$R(m)$$ to find $$\frac{dR}{dm}$$: Now, we differentiate $$R(m)$$ with respect to $$m$$ to find the marginal-revenue product, $$\text{MRP}(m) = \frac{dR}{dm}$$. Let me perform the calculation to find the MRP at $$m = 90$$.The marginal-revenue product (MRP) at $$m = 90$$ is $$-217$$. This negative value indicates that producing additional units by hiring more employees beyond this point would reduce revenue. Would you like more details or have any questions? Here are 5 related questions to explore further: 1. How does marginal-revenue product change as the number of employees increases? 2. What does a negative MRP imply for business decisions? 3. How would the result change if the production function had a different form? 4. How can you determine the optimal number of employees based on MRP? 5. How is MRP related to labor demand in economics? **Tip:** Marginal-revenue product helps businesses decide the optimal level of labor input, balancing production efficiency and costs.