Question

select the correct answer.

mrs. allen, a high school math teacher, is grading quizzes and tests this weekend. she has at most 4 hours, or 240 minutes, to spend grading, and she would like to get at least 60 quizzes and tests graded. on average, it takes her 3 minutes to grade each quiz and 8 minutes to grade each test.

if (x) represents the number of quizzes and (y) represents the number of tests mrs. allen grades this weekend, which point lies in the solution set?

(55,9)
(32,21)
(45,18)
(70,5)

Explanation:

🆕 New Concept Discovered: Systems of Linear Inequalities
Using math to find regions that satisfy multiple limits

Step 1: Define the variables and constraints

Let:

• \( x \) = number of quizzes graded
• \( y \) = number of tests graded

From the problem description, we have two main constraints:

1. Total graded items constraint: Mrs. Allen wants to grade at least 60 items.

$$ x + y \ge 60 $$

1. Time constraint: She has at most 240 minutes. Grading each quiz takes 3 minutes, and each test takes 8 minutes.

$$ 3x + 8y \le 240 $$

Step 2: Test the given options

We need to find which coordinate pair \( (x, y) \) satisfies both inequalities.

Option 1: \( (55, 9) \)

• Total items:

$$ 55 + 9 = 64 \ge 60 \quad \text{(True)} $$

• Total time:

$$ 3(55) + 8(9) = 165 + 72 = 237 \le 240 \quad \text{(True)} $$

Since both conditions are met, the point \( (55, 9) \) lies in the solution set.

Option 2: \( (32, 21) \)

• Total items:

$$ 32 + 21 = 53 \ge 60 \quad \text{(False)} $$

Option 3: \( (45, 18) \)

• Total items:

$$ 45 + 18 = 63 \ge 60 \quad \text{(True)} $$

• Total time:

$$ 3(45) + 8(18) = 135 + 144 = 279 \le 240 \quad \text{(False)} $$

Option 4: \( (70, 5) \)

• Total items:

$$ 70 + 5 = 75 \ge 60 \quad \text{(True)} $$

• Total time:

$$ 3(70) + 8(5) = 210 + 40 = 250 \le 240 \quad \text{(False)} $$

Answer:

\( (55,9) \)