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

Let $x$ = number of footballs, $y$ = number of baseballs.
Profit function: $P = 6x + 5.5y$
Constraints:
$35 \leq x \leq 45$
$40 \leq y \leq 55$
$x + y \leq 80$

Step2: Identify feasible vertices

To maximize profit, test vertices of the feasible region:

1. $x=35, y=45$ (since $35+45=80$)
2. $x=40, y=40$
3. $x=45, y=35$ (but $y$ must be $\geq40$, so invalid)
4. $x=35, y=55$ (but $35+55=90>80$, invalid)
5. $x=45, y=40$ (since $45+40=85>80$, invalid)

Valid vertices: $(35,45), (40,40), (35,40)$

Step3: Calculate profit at each vertex

• For $(35,45)$:

$P = 6(35) + 5.5(45) = 210 + 247.5 = 457.5$

• For $(40,40)$:

$P = 6(40) + 5.5(40) = 240 + 220 = 460$

• For $(35,40)$:

$P = 6(35) + 5.5(40) = 210 + 220 = 430$

Step4: Compare profits

The highest profit is $460.00$

Answer:

$460.00$