Pierce Manufacturing determines that the daily revenue, in dollars, from the sale of $x$ lawn chairs is $R(x)=0.006 x^{3}+0.04 x^{2}+0.3 x$. Currently, Pierce sells 90 lawn chairs daily.
a) What is the current daily revenue?
b) How much would revenue increase if 94 lawn chairs were sold each day?
c) What is the marginal revenue when 90 lawn chairs are sold daily?
d) Use the answer from part (c) to estimate $R(91), R(92)$, and $R(93)$.
Answer
\(\boxed{\text{Final Answer:}}\) a) The current daily revenue is \(\boxed{\$4725}\). b) The revenue would increase by approximately \(\boxed{\$640.14}\) if 94 lawn chairs were sold each day. c) The marginal revenue when 90 lawn chairs are sold daily is \(\boxed{\$153.3}\). d) The estimated revenues for selling 91, 92, and 93 lawn chairs are approximately \(\boxed{\$4878.3}\), \(\boxed{\$5031.6}\), and \(\boxed{\$5184.9}\), respectively.
Step 1 ：Let's denote the number of lawn chairs sold as \(x\) and the revenue function as \(R(x) = 0.006x^{3} + 0.04x^{2} + 0.3x\).
Step 2 ：For part a), we substitute \(x = 90\) into the revenue function to find the current daily revenue: \(R(90) = 0.006(90)^{3} + 0.04(90)^{2} + 0.3(90) = \$4725\).
Step 3 ：For part b), we calculate the revenue when \(x = 94\) and subtract the current daily revenue from it to find the increase in revenue: \(R(94) = 0.006(94)^{3} + 0.04(94)^{2} + 0.3(94) = \$5365.14\). The increase in revenue is \(5365.14 - 4725 = \$640.14\).
Step 4 ：For part c), we find the derivative of the revenue function, \(R'(x) = 0.018x^{2} + 0.08x + 0.3\), and substitute \(x = 90\) into it to find the marginal revenue: \(R'(90) = 0.018(90)^{2} + 0.08(90) + 0.3 = \$153.3\).
Step 5 ：For part d), we use the marginal revenue to estimate the revenue for selling 91, 92, and 93 lawn chairs. The marginal revenue gives the approximate increase in revenue for each additional lawn chair sold, so we add the marginal revenue to the current revenue to estimate the revenue for selling one more lawn chair: \(R(91) = R(90) + R'(90) = \$4878.3\), \(R(92) = R(91) + R'(90) = \$5031.6\), and \(R(93) = R(92) + R'(90) = \$5184.9\).
Step 6 ：\(\boxed{\text{Final Answer:}}\) a) The current daily revenue is \(\boxed{\$4725}\). b) The revenue would increase by approximately \(\boxed{\$640.14}\) if 94 lawn chairs were sold each day. c) The marginal revenue when 90 lawn chairs are sold daily is \(\boxed{\$153.3}\). d) The estimated revenues for selling 91, 92, and 93 lawn chairs are approximately \(\boxed{\$4878.3}\), \(\boxed{\$5031.6}\), and \(\boxed{\$5184.9}\), respectively.
