As manager of Citywide Racquet Club, you must determine the best price to charge for locker rentals. Assume that the (marginal) cost of providing lockers is 0. The monthly demand for lockers is estimated to be Q = 100 - 2P where P is the monthly rental price and Q is the number of lockers rented per month. What price would you charge? How many lockers are rented monthly at this price? Explain why you chose this price.
To determine the best price to charge for locker rentals, we need to analyze the demand equation and find the price that maximizes revenue.
The monthly demand for lockers is given by Q = 100 - 2P, where P is the monthly rental price and Q is the number of lockers rented per month.
To find the price that maximizes revenue, we need to find the price at which the demand equation intersects the revenue equation. Revenue is calculated as the product of price and quantity, R = P * Q.
Substituting the demand equation into the revenue equation, we get:
R = P * (100 - 2P)
R = 100P - 2P^2
To maximize revenue, we take the derivative of the revenue equation with respect to P and set it equal to zero:
dR/dP = 100 - 4P = 0
Solving for P, we find:
100 - 4P = 0
4P = 100
P = 25
Therefore, the price that maximizes revenue is $25 per month.
To determine the number of lockers rented at this price, we substitute P = 25 into the demand equation:
Q = 100 - 2(25)
Q = 100 - 50
Q = 50
So, at a price of $25 per month, 50 lockers are rented monthly.
I chose this price because it is the price that maximizes revenue. It is the point at which the demand equation intersects the revenue equation, resulting in the highest possible revenue for the locker rentals.
To determine the best price to charge for locker rentals, you need to find the price that maximizes your revenue. Revenue is calculated as the product of the price and the quantity sold.
In this case, the monthly demand for lockers is given by the equation Q = 100 - 2P, where Q is the number of lockers rented per month and P is the monthly rental price. We can rearrange this equation to solve for P:
P = (100 - Q) / 2
To maximize revenue, we need to find the value of Q that will yield the highest value for P. One way to do this is to plot the demand curve and find the point where the demand curve intersects the x-axis (P-axis) at the highest possible value.
First, we can set Q = 0 and calculate the corresponding value of P:
P = (100 - 0) / 2 = 50
This tells us that if we charge $50 per month for locker rentals, we will rent out 0 lockers. This is the upper bound on our price because any higher price would result in no demand.
Next, we set P = 0 and calculate the corresponding value of Q:
0 = (100 - Q) / 2
0 = 100 - Q
Q = 100
This tells us that if we charge $0 per month for locker rentals, we will rent out 100 lockers. This is the lower bound on our price because any lower price would result in renting out all available lockers.
Now, with these two points, we can plot the demand curve and determine the price that yields the highest revenue. The demand curve will have a negative slope, starting at (P = $50, Q = 0) and ending at (P = $0, Q = 100). We can draw a straight line connecting these two points to represent the demand curve.
To find the price that maximizes revenue, we need to find the point where the demand curve intersects the x-axis (P-axis) at the highest possible value. This point corresponds to the price at which we can rent out the most lockers.
By visually inspecting the demand curve, we can see that this occurs at a price of $25, where Q = 50.
Therefore, I would suggest charging $25 per month for locker rentals, as this price would result in 50 lockers being rented monthly. This choice maximizes revenue by finding the balance between price and demand, allowing for a larger number of lockers to be rented while still generating a significant amount of income.
You want to maximize income which is P*Q
P Q = 100 P - 2 P ^2
2 p^2 -100 p = - income
where is that derivative = 0?
4 p - 100
p = 25 seems to be it.
then you rent
Q = 100 - 50 = 50 of them
and you make
PQ = $1250
If you have not had calculus, find the vertex of that parabola by completing the square