the profits in hundreds of dollars, $p(c)$, that a company can make from a product is modeled by a function…
the profits in hundreds of dollars, $p(c)$, that a company can make from a product is modeled by a function of the price, $c$, they charge for the product: $p(c)= - 20c^{2}+320c + 5120$. what is the maximum profit the company can make from the product?\n$540,000\n$640,000\n$800,000\n$896,000
Answer
Explanation:
Step1: Identify the function type
The profit function $P(c)=- 20c^{2}+320c + 5120$ is a quadratic function in the form $y = ax^{2}+bx + c$ where $a=-20$, $b = 320$, and $c = 5120$. Since $a<0$, the parabola opens down - ward and the vertex of the parabola gives the maximum value of the function.
Step2: Find the x - coordinate of the vertex
The x - coordinate of the vertex of a quadratic function $y = ax^{2}+bx + c$ is given by $x=-\frac{b}{2a}$. For $P(c)=-20c^{2}+320c + 5120$, we have $c=-\frac{320}{2\times(-20)}=\frac{-320}{-40}=8$.
Step3: Find the maximum profit
Substitute $c = 8$ into the profit function $P(c)$: $P(8)=-20\times8^{2}+320\times8 + 5120$. First, calculate $-20\times8^{2}=-20\times64=-1280$. Then, calculate $320\times8 = 2560$. $P(8)=-1280 + 2560+5120$. $P(8)=-1280+7680=6400$. Since $P(c)$ is in hundreds of dollars, the maximum profit is $6400\times100=$640000$.
Answer:
$640000$ (corresponding to the option: $$640,000$)