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 $R(93)$.
a) The current revenue is $\$ 4725$.
b) The revenue would increase by $\$ 640.14$. (Round to the nearest cent.)
c) The marginal revenue is when 90 lawn chairs are sold daily.
Answer
Final Answer: a) The current daily revenue is \(\boxed{4725}\) dollars. b) The revenue would increase by \(\boxed{640.14}\) dollars if 94 lawn chairs were sold each day. c) The marginal revenue when 90 lawn chairs are sold daily is \(\boxed{153.3}\) dollars per chair.
Step 1 ：The revenue function, R(x), is given by \(R(x)=0.006 x^{3}+0.04 x^{2}+0.3 x\). This function gives the total revenue from selling x units of a product, in this case, lawn chairs.
Step 2 ：To find the current daily revenue, we substitute the current daily sales (90 lawn chairs) into the revenue function. This gives us \(R(90)=0.006(90)^{3}+0.04(90)^{2}+0.3(90)\), which simplifies to \$4725.
Step 3 ：To find how much the revenue would increase if 94 lawn chairs were sold each day, we calculate the revenue for 94 lawn chairs and subtract the current daily revenue from it. This gives us \(R(94)=0.006(94)^{3}+0.04(94)^{2}+0.3(94)\), which simplifies to \$5365.14. Subtracting the current revenue from this gives us an increase of \$640.14.
Step 4 ：The marginal revenue is the derivative of the revenue function. It gives the rate of change of revenue with respect to the number of units sold. The derivative of the revenue function is \(R'(x)=0.018x^{2}+0.08x+0.3\).
Step 5 ：To find the marginal revenue when 90 lawn chairs are sold daily, we substitute 90 into the derivative of the revenue function. This gives us \(R'(90)=0.018(90)^{2}+0.08(90)+0.3\), which simplifies to \$153.3.
Step 6 ：Final Answer: a) The current daily revenue is \(\boxed{4725}\) dollars. b) The revenue would increase by \(\boxed{640.14}\) dollars if 94 lawn chairs were sold each day. c) The marginal revenue when 90 lawn chairs are sold daily is \(\boxed{153.3}\) dollars per chair.
