Marginal Revenue 13E5Bb

1. **State the problem:** We are given the revenue function $$R(x)=41x-\frac{x^2}{250}$$ where $x$ is the number of picnic tables sold. 2. **Find the marginal revenue $R'(x)$:** The marginal revenue is the derivative of the revenue function with respect to $x$. $$R'(x) = \frac{d}{dx}\left(41x - \frac{x^2}{250}\right) = 41 - \frac{2x}{250} = 41 - \frac{x}{125}$$ 3. **Estimate the revenue from the sale of the 3001st table by finding $R'(3000)$:** Substitute $x=3000$ into the marginal revenue formula: $$R'(3000) = 41 - \frac{3000}{125} = 41 - 24 = 17$$ So, the estimated revenue from the 3001st table is $17$. 4. **Determine the actual revenue from the sale of the 3001st table:** The actual revenue from the 3001st table is the difference between the revenue at $x=3001$ and $x=3000$: $$R(3001) = 41(3001) - \frac{3001^2}{250}$$ $$R(3000) = 41(3000) - \frac{3000^2}{250}$$ Calculate the difference: $$R(3001) - R(3000) = \left(41(3001) - \frac{3001^2}{250}\right) - \left(41(3000) - \frac{3000^2}{250}\right)$$ Simplify step-by-step: $$= 41(3001 - 3000) - \frac{3001^2 - 3000^2}{250} = 41 - \frac{(3001 - 3000)(3001 + 3000)}{250}$$ $$= 41 - \frac{1 \times 6001}{250} = 41 - 24.004 = 16.996 \approx 17.00$$ 5. **Compare the answers for parts (b) and (c):** The marginal revenue estimate ($17$) is very close to the actual revenue ($17.00$) from the sale of the 3001st table. **Conclusion:** The marginal revenue provides a good approximation of the revenue from the sale of the 3001st table because the values are nearly identical.