lAURA OWNS A PIE COMPANY. SHE HAS LEARNED THAT HER PROFITS, P(X), FROM THE SALE OF X CASES OF PIES ARE GIVEN BY P(X)= 150X - X^2.
(A) HOW MANY CASES OF PIES SHOULD SHE SELL IN ORDER TO MAXIMIZE PROFIT?
(B) WHAT IS THE MAXIMUM PROFIT?
To find the number of cases of pies Laura should sell to maximize profit, we need to determine the value of X that gives the maximum value of P(X). This can be done by finding the vertex of the parabolic function P(X) = 150X - X^2.
(A) To find the number of cases that will maximize profit:
1. Identify the coefficient of the quadratic term (X^2) in the equation P(X) = 150X - X^2. In this case, the coefficient is -1.
2. Use the formula for the X-coordinate of the vertex, which is given by -b/2a, where a is the coefficient of the quadratic term and b is the coefficient of the linear term. In this case, a = -1 and b = 150.
X-coordinate of the vertex = -b/2a = -150/(2*-1) = 75.
3. Therefore, Laura should sell 75 cases of pies in order to maximize profit.
(B) To find the maximum profit:
1. Substitute the value of X (which is 75) into the profit function P(X) = 150X - X^2.
P(75) = 150 * 75 - 75^2 = 11250 - 5625 = 5625.
2. The maximum profit Laura can make is $5625.