A manufacturer has found that when m employees are working, the number of units of product per day is $q=10\sqrt{m^2+4900}-700$. The demand equation for the product is $8q + p^2 -19300 = 0$, where $p$ is the selling price when the demand for the product is $q$ units per day.
Determine the manufacturer's marginal-revenue product when $m = 240$
Find the relative rate of change of revenue with respect to number of employees when $m = 240$
Suppose it would cost the manufacturer $400$ more per day to hire an additional employee. Would you hire the $241$ st employee, why or why not?
Notes: Marginal-revenue product formula is Marginal Revenue $\cdot$ Marginal Product. Marginal is another way of saying rate of change so it is the derivative. And revenue $= p \cdot q$.
So, I solved all three of them but I just want another eye to check it - it is kinda important.
Here is the solution with proper form:
1.$\text{MRP} = \text{MR} \cdot \text{MP}$ ,$p = \sqrt{19300-8q}$
On substituting $m=240$ into $q$ product function: $q=10\sqrt{240^2+4900}-700 = 1800$
$$ \begin{align} \text{R} &= p \cdot q \\ &= \sqrt{19300-8q} \cdot q\\ &= \sqrt{19300q^2-8q^3} \end{align}$$
$$\text{MR} = \text{R'} = \frac{19300q-12q^2}{\sqrt{19300q^2-8q^3}}$$ On substituting $q = 1800$ $$ \text{MR} = -32.86$$
Then to find the $\text{MP}$,$$ \text{MP} = q' = \frac{10m}{\sqrt{m^2+4900}}$$ On substituting $m = 240$ $$\text{MP} = 9.6$$
Then $$\text{MRP} = \text{MR} \cdot \text{ MP} = -32.86 \cdot 9.6 = -315.43$$
2.Relative Rate of change is $\frac{ \text{MR}}{\text{R}}$, and for $m = 240$, I found $p=70$ and $q=1800$. Then $$\text{R} = p \cdot q = 70 \cdot 1800 = 126000$$ $$ \implies \frac{\text{MR}}{\text{R}} = \frac{-32.86}{126000} = -0.00026 $$
Especially I have doubts about the second part of the question. When it says find the relative rate of change with respect to m do I need to use $\frac{dR}{dm} * \frac{dq}{dm}$ or what I did is fine?
- I wouldn't advise since the marginal revenue is negative which means the $241$st employee will bring negative revenue.