Use P= -7200+300x-2x^2
p= profit
x- selling price
question: Determine the selling price when 90% of the maximum possible profit is earned
To determine the selling price when 90% of the maximum possible profit is earned, we need to find the maximum profit first.
The equation given is P = -7200 + 300x - 2x^2, where P represents the profit and x represents the selling price.
To find the maximum profit, we can use the concept of the vertex of a quadratic function. The vertex can be found using the formula x = -b / (2a) for a quadratic equation in the form ax^2 + bx + c = 0.
In this case, the equation is -2x^2 + 300x - 7200 = 0, so a = -2, b = 300, and c = -7200.
Applying the formula, we get x = -300 / (2 * -2) = -300 / -4 = 75.
Therefore, the selling price (x) at which the maximum profit is earned is 75.
Now, to find the profit at this selling price, substitute x = 75 into the equation P = -7200 + 300x - 2x^2:
P = -7200 + 300(75) - 2(75)^2
P = -7200 + 22500 - 2(5625)
P = -7200 + 22500 - 11250
P = 4150
Hence, the maximum profit is 4150.
To determine the selling price when 90% of the maximum possible profit is earned, we simply need to multiply the maximum profit by 0.9:
90% of 4150 = 0.9 * 4150 = 3735
Therefore, the selling price when 90% of the maximum possible profit is earned is $3735.