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 36 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.
[tex]\begin{array}{l}
3 w+6 r \geq 36 \\
3 w+6 r \leq 90
\end{array}[/tex]
Which combination of hours can Keitaro walk and run in a month to reach his goal?
A. 2 hours walking; 12 hours running
B. 4 hours walking; 3 hours running
C. 9 hours walking; 12 hours running
D. 12 hours walking; 10 hours running
Answer (2)
Keitaro can walk for 2 hours and run for 12 hours to meet his monthly distance goal. This combination satisfies both distance inequalities. Therefore, the correct answer is A. 2 hours walking; 12 hours running.
;
The problem provides two inequalities: 3 w + 6 r ≥ 36 and 3 w + 6 r ≤ 90 .
Test each combination of walking hours ( w ) and running hours ( r ) against the inequalities.
For (2, 12): 3 ( 2 ) + 6 ( 12 ) = 78 , which satisfies 36 ≤ 78 ≤ 90 .
The combination that satisfies both inequalities is 2 hours walking; 12 hours running .
Explanation
Understanding the Problem We are given two inequalities that describe the constraints on the number of hours Keitaro can walk ( w ) and run ( r ) each month:
3 w + 6 r ≥ 36 3 w + 6 r ≤ 90
We need to check which of the given combinations of ( w , r ) satisfy both inequalities.
Testing the Combinations Let's test each combination:
2 hours walking; 12 hours running: ( w , r ) = ( 2 , 12 ) 3 ( 2 ) + 6 ( 12 ) = 6 + 72 = 78 . Since 36 ≤ 78 ≤ 90 , this combination satisfies both inequalities.
4 hours walking; 3 hours running: ( w , r ) = ( 4 , 3 ) 3 ( 4 ) + 6 ( 3 ) = 12 + 18 = 30 . Since 30 < 36 , this combination does not satisfy the first inequality.
9 hours walking; 12 hours running: ( w , r ) = ( 9 , 12 ) 3 ( 9 ) + 6 ( 12 ) = 27 + 72 = 99 . Since 90"> 99 > 90 , this combination does not satisfy the second inequality.
12 hours walking; 10 hours running: ( w , r ) = ( 12 , 10 ) 3 ( 12 ) + 6 ( 10 ) = 36 + 60 = 96 . Since 90"> 96 > 90 , this combination does not satisfy the second inequality.
Finding the Solution Only the combination of 2 hours walking and 12 hours running satisfies both inequalities.
Examples
This type of problem is useful in resource allocation. For example, a farmer might want to determine how many acres of two different crops to plant, given constraints on land, water, and budget. Similarly, a factory manager might want to decide how many units of two different products to manufacture, given constraints on labor, materials, and machine time. By setting up a system of inequalities, one can find the feasible combinations that satisfy all the constraints.
