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 form of the function
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 $c = 5120$.
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 $c=-\frac{b}{2a}$. Substituting $a=-20$ and $b = 320$ into the formula, we get $c=-\frac{320}{2\times(-20)}=\frac{-320}{-40}=8$.
Step3: Find the maximum value of the function
Substitute $c = 8$ into the profit function $P(c)=-20c^{2}+320c + 5120$. Then $P(8)=-20\times8^{2}+320\times8 + 5120$. First, calculate $-20\times8^{2}=-20\times64=-1280$, and $320\times8 = 2560$. So $P(8)=-1280+2560 + 5120=6400$. Since the profit function $P(c)$ is in hundreds of dollars, the maximum profit is $6400\times100=$640000$.
Answer:
$640,000$ (corresponding to the second option)