Question

keitaro walks at a pace of 3 miles per hour and runs at a pace of 6 miles per hour. each month, he wants to complete at least 30 miles but not more than 90 miles. the system of inequalities represents the number of hours he can walk, \\(w\\), and the number of hours he can run, \\(r\\), to reach his goal.

\\3w + 6r \ge 30\\
\\3w + 6r \le 90\\

which combination of hours can keitaro walk and run in a month to reach his goal?

• 4 hours walking, 3 hours running
• 2 hours walking, 12 hours running
• 9 hours walking, 12 hours running
• 12 hours walking, 10 hours running

Explanation:

Identify the system of inequalities

Using the Systems of Linear Inequalities knowledge point

$$
LATEXBLOCK0
$$

Test the given options

Using the Constraint Verification knowledge point

• Option A: \(w = 4, r = 3\)

$$ 3(4) + 6(3) = 12 + 18 = 30 \quad (\text{fails } \ge 36) $$

• Option B: \(w = 2, r = 12\)

$$ 3(2) + 6(12) = 6 + 72 = 78 \quad (36 \le 78 \le 90) $$

• Option C: \(w = 9, r = 12\)

$$ 3(9) + 6(12) = 27 + 72 = 99 \quad (\text{fails } \le 90) $$

• Option D: \(w = 12, r = 10\)

$$ 3(12) + 6(10) = 36 + 60 = 96 \quad (\text{fails } \le 90) $$

Confirm the correct combination

Using the Constraint Verification knowledge point

$$ w = 2, r = 12 \implies 36 \le 78 \le 90 $$

Answer:

• (A) 4 hours walking; 3 hours running
• (B) 2 hours walking; 12 hours running (Correct answer)
• (C) 9 hours walking; 12 hours running
• (D) 12 hours walking; 10 hours running