Question

a sports memorabilia store makes $6 profit on each football it sells and $5.50 profit on each baseball it sells. in a typical month, it sells between 35 and 45 footballs and between 40 and 55 baseballs. the store can stock no more than 80 balls total during a single month. what is the maximum profit the store can make from selling footballs and baseballs in a typical month?
$457.50
$460.00
$462.50
$572.50

Explanation:

Step1: Define variables and objective

Let $x$ = number of footballs, $y$ = number of baseballs.
Profit function: $P = 6x + 5.5y$

Step2: List constraints

1. $35 \leq x \leq 45$
2. $40 \leq y \leq 55$
3. $x + y \leq 80$

Step3: Maximize profit strategy

Since footballs have higher profit ($6 > 5.5$), prioritize maximum $x$ first, then find valid $y$.
Max $x = 45$. Then $y \leq 80 - 45 = 35$, but $y \geq 40$ (invalid).
Next, reduce $x$ to meet $y \geq 40$:
$x = 80 - y$. To satisfy $y \geq 40$, $x \leq 40$. Max $x$ here is 40.

Step4: Calculate corresponding y

$y = 80 - 40 = 40$ (meets $40 \leq y \leq 55$)

Step5: Compute total profit

$P = 6(40) + 5.5(40) = 240 + 220 = 460$
Verify next possible: $x=39$, $y=41$: $P=6(39)+5.5(41)=234+225.5=459.5 < 460$

Answer:

$460.00$ (Option B)