To make pom poms in our school colors, we will have expenses of $100 to rent the Acme PomPom Plant, and then $0.25 per pom pom in materials. We believe that we can sell 500 pom poms if we charge $1.50. Assume the number sold is a linear function of the price. How much should we charge to maximize profit?
I have done this problem many times but I cannot find max profit. I keep fining a max income. I am lost and don't know what to do please help!
"Assume the number sold is a linear function of the price"
but no information is given on the price elasticity, so we will assume, in general, an increase in price of $1 will increase sales by m units.
Under normal supply-demand curve, m is necessarily negative, of the order of -200 or so.
With that in mind, and knowing that (500,1.50) is a point on the line sales versus price, we construct the sales(y)-price(x) relation as:
(y-500)=m(x-1.50)
therefore, at a price of x, we expect sales of
y=m(x-1.5)+500
Total revenue,
R=xy
Total profit
P=xy - cost
=xy - (0.25x+100)
=x(m(x-1.5)+500) - (0.25x+100)
To get the maximum profit, we differentiate profit with respect to price, and equate to zero to find the optimum price:
dp/dx = m*x+m*(x-1.5)+1999/4 =0
Solve for x:
x=(6*m-1999)/(8*m)
If the price elasticity m=-100,
x=2599/800=$3.25
if m=-200
x=3199/1600=$2
if m=-300
x=3799/2400=$1.58
...
etc.
To find the price that will maximize profit, we need to understand the relationship between the price and the number of pom poms sold, as well as the relationship between the price, expenses, and revenue.
Let's start by defining some variables:
P = price per pom pom
N = number of pom poms sold
E = expenses
R = revenue
P = profit
Given:
Expenses (E) = $100 to rent the Acme PomPom Plant
Materials cost (M) = $0.25 per pom pom
Price (P) = $1.50
Number of pom poms sold (N) = 500
We can start by calculating the total revenue (R) generated by selling pom poms:
Revenue (R) = Price per pom pom (P) * Number of pom poms sold (N)
R = P * N
In this case, R = $1.50 * 500 = $750
Next, we can calculate the total expenses (E) incurred for renting the Acme PomPom Plant and purchasing materials for the pom poms:
Expenses (E) = $100 (rent) + Materials cost per pom pom (M) * Number of pom poms sold (N)
E = $100 + $0.25 * 500 = $100 + $125 = $225
Now we can calculate the profit (P) by subtracting the expenses from the revenue:
Profit (P) = Revenue (R) - Expenses (E)
P = R - E
P = $750 - $225 = $525
Now that we have derived the profit equation, we can determine how much you should charge for a pom pom to maximize profit. To find the maximum profit, we need to find the maximum point on the profit curve. In this case, since the number of pom poms sold is a linear function of the price, we can plot the profit equation on a graph.
Let's assume that the price is on the X-axis and the profit is on the Y-axis. We have already calculated the profit (P) for one point ($525).
To find the maximum profit, we need to determine the price (P) at which the profit (Y) will be maximized. One way to do this is by calculating the slope of the profit equation.
Slope = change in profit (ΔP) / change in price (ΔX)
Slope = ($525 - $0) / (X - $1.50)
To maximize the profit, we want the slope to be zero. Setting the slope equation to zero and solving for X will give us the price that maximizes the profit.
0 = ($525 - $0) / (X - $1.50)
Now we can solve for X:
0 = $525 / (X - $1.50)
Cross-multiplying gives us:
0 * (X - $1.50) = $525
0 = $525
Since we have a division by zero, there is no defined price that will maximize the profit.
However, it is worth noting that the profit will be maximized when the price is set at a level that maximizes the number of pom poms sold, given the cost of the materials and the rental expenses. In this case, that price is $1.50, which gives a profit of $525.
So, to maximize profit in this scenario, you should charge $1.50 per pom pom.