Pierce Manufacturing determines that the daily revenue, in dollars, from the sale of $x$ lawn chairs is $R(x)=0.007 x^{3}+0.03 x^{2}+0.3 x$. Currently, Pierce sells 80 lawn chairs daily. a) What is the current daily revenue? b) How much would revenue increase if 82 lawn chairs were sold each day? c) What is the marginal revenue when 80 lawn chairs are sold daily? d) Use the answer from part (c) to estimate $R(81), R(82)$, and $R(83)$. a) The current revenue is $\$ 3800$. b) The revenue would increase by $\$ 285.9$. (Round to the nearest cent.) c) The marginal revenue is $\$ 139.5$ when 80 lawn chairs are sold daily. d) $\mathrm{R}(81) \approx \$ \square$ (Round to the nearest cent as needed. Use the answer from part $c$ to answer this part.)
Solution
Step 1 ：The current daily revenue is \(\$ 3800\).
Step 2 ：The revenue would increase by \(\$ 285.9\) if 82 lawn chairs were sold each day.
Step 3 ：The marginal revenue when 80 lawn chairs are sold daily is \(\$ 139.5\).
Step 4 ：We can use the marginal revenue to estimate the revenue when 81 lawn chairs are sold daily. The marginal revenue is the rate of change of the revenue with respect to the number of lawn chairs sold. So, we can estimate the revenue for 81 chairs by adding the marginal revenue to the revenue for 80 chairs.
Step 5 ：revenue_80 = 3800
Step 6 ：marginal_revenue_80 = 139.5
Step 7 ：revenue_81 = revenue_80 + marginal_revenue_80
Step 8 ：\(\boxed{\text{The estimated revenue when 81 lawn chairs are sold daily is approximately \$3939.50.}}\)
