Answer the problem.
-An approximation to the total profit (in thousands of dollars) from the sale of $x$ hundred thousand tires is given by $p = - x ^ { 3 } + 15 x ^ { 2 } - 48 x + 450 , x \geq 3$. Find the number of hundred thousands of tires that must be sold to maximize profit.
A) 8 hundred thousand
B) 10 hundred thousand
C) 3 hundred thousand
D) 5 hundred thousand

Question

Quizplus · Accepted Answer

To maximize profit, we need to find the critical points of the profit function $p = -x^3 + 15x^2 - 48x + 450$. We do this by taking the derivative of $p$ with respect to $x$ and setting it equal to zero. The derivative is $p' = -3x^2 + 30x - 48$. Setting $p'$ equal to zero gives the quadratic equation $-3x^2 + 30x - 48 = 0$. Solving this equation for $x$ gives the critical points. The critical point that falls within the domain $x \geq 3$ and maximizes the profit is $x = 8$, which corresponds to selling 8 hundred thousand tires.

Answer the Problem xxx Hundred Thousand Tires Is Given By p=−x3+15x2−48x+450,x≥3p = - x ^ { 3 } + 15 x ^ { 2 } - 48 x + 450 , x \geq 3p=−x3+15x2−48x+450,x≥3

Answer the Problem $x$ Hundred Thousand Tires Is Given By $p = - x ^ { 3 } + 15 x ^ { 2 } - 48 x + 450 , x \geq 3$