Question

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brought 300 hot dogs and 800 hamburgers to sell.
the graph shows the feasible region, where ( x ) represents the number of hot dogs sold and ( y ) represents the number of hamburgers sold.
which ordered pairs meet all the constraints for a successful fundraiser and make sense in context of the situation?
select each correct answer.
( (0, 350) )
( (250, 700) )
( (300, 350) )

Explanation:

Step1: Identify constraints from context

• $0 \leq x \leq 300$ (max 300 hot dogs)
• $0 \leq y \leq 800$ (max 800 hamburgers)
• Feasible region: below $y = -\frac{1}{2}x + 350$, above $y = 2x$

Step2: Check (0, 350)

$x=0 \leq 300$, $y=350 \leq 800$; $350 = -\frac{1}{2}(0)+350$, $350 \geq 2(0)$. Valid.

Step3: Check (250, 700)

$y=700 > 2(250)=500$, but $700 > -\frac{1}{2}(250)+350=225$. Lies outside feasible region. Invalid.

Step4: Check (200, 350)

$x=200 \leq 300$, $y=350 \leq 800$; $350 \geq 2(200)=400$? No, wait $350 \leq -\frac{1}{2}(200)+350=250$? No, correction: $y \leq -\frac{1}{2}x + 350$? No, the feasible region is above $y=2x$? No, looking at the graph: the lower line is $y=2x$ (from (0,0) to (150,300)), upper left is (0,350), upper right is (300,600). Wait no, the feasible region is the shaded area: bounded by $x \leq 300$, $y \leq 800$, $y \geq -\frac{1}{2}x + 350$? No, (0,350) is a vertex, (150,250) is a vertex, (300,600) is a vertex. So the constraints are:
$0 \leq x \leq 300$
$0 \leq y \leq 800$
$y \geq -\frac{2}{3}x + 350$ (wait (150,250): $250 = -\frac{2}{3}(150)+350= -100+350=250$)
$y \leq 2x$ (300,600: 600=2*300)
Now recheck:
(0,350): $350 \geq -\frac{2}{3}(0)+350=350$, $350 \leq 2(0)=0$? No, wait the shaded area is above the line from (0,350) to (150,250) and below the line from (0,0) to (300,600), and left of x=300, below y=800. So (0,350) is on the boundary, valid.
(250,700): 700 > 2*250=500, so above the upper line, outside shaded area. Invalid.
(200,350): 350 ≤ 2200=400, and 350 ≥ -2/3200 +350 ≈ 350-133.33=216.67. So 350 is between 216.67 and 400, and x=200 ≤300, y=350 ≤800. So it is inside the shaded region. Valid.

Step5: Confirm valid pairs

Final valid pairs: (0,350), (200,350)

Answer:

(0, 350), (200, 350)