If \( x \) is the number of units produced and \( n \) is the number of workers, such that \( x=\displaystyle\frac{8(2n+1)}{\sqrt{n^2+225}} \). The demand function is given by \( p=\displaystyle\frac{845}{x+5} \), then the marginal revenue for \( n=8 \) is :
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 25
When n = 8, the output is
\( \displaystyle x=\frac{8(2\times 8+1)}{\sqrt{64+225}} \)
\( \displaystyle =\frac{8\times 17}{\sqrt{289}}=\frac{136}{17}=8 \)
Revenue is price × quantity:
\( \displaystyle R=px=\frac{845x}{x+5} \)
Marginal revenue is the derivative of R with respect to x:
\( \displaystyle \mathrm{MR}=\frac{dR}{dx}=\frac{845(x+5)-845x}{(x+5)^2} \)
\( \displaystyle =\frac{4225}{(x+5)^2} \)
At x = 8:
\( \displaystyle \mathrm{MR}=\frac{4225}{13^2} \)
\( \displaystyle =\frac{4225}{169}=25 \)
Why the other options are incorrect
Option (1) 65 — Incorrect. 65 is the price (average revenue) at x = 8, \( \frac{845}{13} \), not the marginal revenue.
Option (2) 13 — Incorrect. 13 is the value of x + 5, not the marginal revenue.
Option (4) 75/13 — Incorrect. The marginal revenue at x = 8 is 25, not \( \frac{75}{13} \).