The daily profit in dollars made by an automobile manufacturer is $P ( x ) = - 45 x ^ { 2 } + 2,250 x - 18,000$ where x is the number of cars produced per shift. Find the maximum possible daily profit.
A) $8,075
B) $9,448
C) $10,125
D) $15,275

Question

Quizplus · Accepted Answer

The maximum profit can be found by using the vertex formula for a parabola, where the x-coordinate of the vertex is given by $-\frac{b}{2a}$ in a quadratic equation of the form $ax^2 + bx + c$. Here, $a = -45$, $b = 2250$, and $c = -18000$. Plugging these values into the formula gives the number of cars produced for maximum profit: $x = -\frac{2250}{2(-45)} = 25$. Substituting $x = 25$ back into the profit equation gives the maximum daily profit: $P(25) = -45(25)^2 + 2250(25) - 18000 = 10125$.

The Daily Profit in Dollars Made by an Automobile Manufacturer P(x)=−45x2+2,250x−18,000P ( x ) = - 45 x ^ { 2 } + 2,250 x - 18,000P(x)=−45x2+2,250x−18,000

The Daily Profit in Dollars Made by an Automobile Manufacturer $P ( x ) = - 45 x ^ { 2 } + 2,250 x - 18,000$