Problem
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 $\$ \square$. (Round to the nearest cent.)
Solution
Step 1 ：Substitute $x=80$ into the revenue function $R(x)=0.007 x^{3}+0.03 x^{2}+0.3 x$ to find the current daily revenue. The current revenue is \(\$ 3800\).
Step 2 ：Calculate the revenue when $x=82$ by substituting $x=82$ into the revenue function. The revenue when 82 lawn chairs are sold is \(\$ 4085.90\).
Step 3 ：Subtract the current revenue from the revenue when 82 lawn chairs are sold to find the increase in revenue. The increase in revenue is \(\$ 285.90\).
Step 4 ：\(\boxed{\$285.90}\) is the increase in revenue if 82 lawn chairs were sold each day.
