Demand for pools. Tropical Pools sells an above ground model for p dollars each. The monthly revenue for this model is given by the formula
R (p) = - 0.08p2 + 300p.
Revenue is the product of the price p and the demand (quantity sold).
a) Factor out the price on the right-hand side of the formula.
b) Write a formula D (p) for the monthly demand.
c) Find D (3000).
d) Use the accompanying graph to estimate the price at which the revenue is maximized. Approximately how many pools will be sold monthly at this price?
e) What is the approximate maximum revenue?
f) Use the accompanying graph to estimate the price at which the revenue is zero.
a) To factor out the price on the right-hand side of the formula, we can rewrite the formula as:
R(p) = p(-0.08p + 300)
b) Now, to write a formula for monthly demand, we need to divide revenue by the price:
D(p) = R(p)/p
D(p) = (-0.08p^2 + 300p)/p
D(p) = -0.08p + 300
c) To find D(3000), substitute 3000 for p in the formula for demand:
D(3000) = -0.08(3000) + 300
D(3000) = -240 + 300
D(3000) = 60
So, at a price of $3000, the monthly demand would be 60 pools.
d) To estimate the price at which revenue is maximized, we need to look for the vertex of the parabolic revenue function. The vertex occurs at the price that gives the maximum revenue.
From the given formula, we can see that revenue is a quadratic function of p with a negative coefficient. This means it opens downwards and has a maximum point.
Based on the graph, estimate the price at which the revenue is maximized.
e) To estimate the approximate maximum revenue, we need to find the value of revenue at the price found in the previous step. Substitute this price into the formula for revenue:
R(p) = -0.08p^2 + 300p
Calculate R(p) for the estimated price, and that will give you the approximate maximum revenue.
f) To estimate the price at which the revenue is zero, we need to find the x-intercepts of the graph. These are the points where the revenue is equal to zero. From the graph, approximate the price at which the revenue is zero.