For the equations​ below, q is the total number of units produced per day by m employees of a​ manufacturer, and p is the price per unit at which the q units are sold. Find the​ marginal-revenue product for the given value of m.
qequalsStartFraction 200 m minus m squared Over 20 EndFraction
​, pequalsnegative 0.3 q plus 80​; m equals 90

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:
\[
p(m) = -0.3 \cdot \frac{200m - m^2}{20} + 80 = \frac{-0.3(200m - m^2)}{20} + 80
= \frac{-60m + 0.3m^2}{20} + 80 = \frac{-3m + 0.015m^2}{1} + 80
\]
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.

For the equations below, q is the total number of units produced per day by m employees of a manufacturer, and p is the price per unit at which the q units are sold. Find the marginal-revenue product for the given value of m. q = (200m - m^2) / 20, p = -0.3q + 80; m = 90.

Learn how to calculate the marginal-revenue product (MRP) for a given production and price function. This step-by-step solution demonstrates how to differentiate revenue functions with respect to labor input and analyze the impact on total revenue. Suitable for college-level students.

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