a store uses the expression -2p + 50 to model the number of backpacks it sells per day, where the price, p…

a store uses the expression -2p + 50 to model the number of backpacks it sells per day, where the price, p, can be anywhere from $9 to $15. which price gives the store the maximum amount of revenue, and what is the maximum revenue? (revenue = price x number of backpacks.)\n$9.00 per backpack gives the maximum revenue; the maximum revenue is $32.00.\n$12.00 per backpack gives the maximum revenue; the maximum revenue is $312.00.\n$12.50 per backpack gives the maximum revenue; the maximum revenue is $312.50.\n$15.00 per backpack gives the maximum revenue; the maximum revenue is $20.00.

a store uses the expression -2p + 50 to model the number of backpacks it sells per day, where the price, p, can be anywhere from $9 to $15. which price gives the store the maximum amount of revenue, and what is the maximum revenue? (revenue = price x number of backpacks.)\n$9.00 per backpack gives the maximum revenue; the maximum revenue is $32.00.\n$12.00 per backpack gives the maximum revenue; the maximum revenue is $312.00.\n$12.50 per backpack gives the maximum revenue; the maximum revenue is $312.50.\n$15.00 per backpack gives the maximum revenue; the maximum revenue is $20.00.

Answer

Explanation:

Step1: Define revenue function

Let $R(p)$ be the revenue function. Given number of backpacks sold is $n=-2p + 50$ and revenue $R(p)=p\times n=p(-2p + 50)=-2p^{2}+50p$.

Step2: Find the vertex of the quadratic - function

For a quadratic function $y = ax^{2}+bx + c$, the $x$ - coordinate of the vertex is $x=-\frac{b}{2a}$. In $R(p)=-2p^{2}+50p$, $a=-2$ and $b = 50$. So $p=-\frac{50}{2\times(-2)}=\frac{50}{4}=12.5$.

Step3: Calculate the maximum revenue

Substitute $p = 12.5$ into $R(p)=-2p^{2}+50p$. $R(12.5)=-2\times(12.5)^{2}+50\times12.5=-2\times156.25 + 625=-312.5+625 = 312.5$.

Answer:

$12.50$ per backpack gives the maximum revenue; the maximum revenue is $$312.50$.