Joshua wants to burn at least 400 calories per day, but no more than 600. He does this by walking and playing basketball. Assuming he burns 4 calories per minute walking, [tex]$w$[/tex], and 5 calories per minute spent playing basketball, [tex]$b$[/tex], the situation can be modeled using these inequalities:
[tex]
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Question

Joshua wants to burn at least 400 calories per day, but no more than 600. He does this by walking and playing basketball. Assuming he burns 4 calories per minute walking, [tex]$w$[/tex], and 5 calories per minute spent playing basketball, [tex]$b$[/tex], the situation can be modeled using these inequalities:
[tex] 
\begin{array}{l}<
4 w+5 b \geq 400 <
4 w+5 b \leq 600
\end{array}<
[/tex]
Which are possible solutions for the number of minutes Joshua can participate in each activity? Check all that apply.

A. 40 minutes walking, 40 minutes basketball
B. 60 minutes walking, 20 minutes basketball
C. 20 minutes walking, 60 minutes basketball
D. 50 minutes walking, 50 minutes basketball
E. 60 minutes walking, 80 minutes basketball
F. 70 minutes walking, 60 minutes basketball

GinnyAnswer · Answer

Substitute the values of w and b for each option into the expression 4 w + 5 b .
Check if the result satisfies both inequalities: 400 ≤ 4 w + 5 b ≤ 600 .
If both inequalities are satisfied, then the (w, b) pair is a possible solution.
The possible solutions are 50 minutes walking, 50 minutes basketball and 70 minutes walking, 60 minutes basketball. 50 minutes walking, 50 minutes basketball and 70 minutes walking, 60 minutes basketball

Explanation

Checking the inequalities First, we need to check each option to see if it satisfies the given inequalities:

4 w + 5 b ≥ 400 4 w + 5 b ≤ 600
We will substitute the values of w and b for each option and see if the resulting calorie burn falls between 400 and 600, inclusive.

Evaluating Option 1 Option 1: 40 minutes walking, 40 minutes basketball w = 40 , b = 40 4 ( 40 ) + 5 ( 40 ) = 160 + 200 = 360 Since 360 < 400 , this option does not satisfy the first inequality.

Evaluating Option 2 Option 2: 60 minutes walking, 20 minutes basketball w = 60 , b = 20 4 ( 60 ) + 5 ( 20 ) = 240 + 100 = 340 Since 340 < 400 , this option does not satisfy the first inequality.

Evaluating Option 3 Option 3: 20 minutes walking, 60 minutes basketball w = 20 , b = 60 4 ( 20 ) + 5 ( 60 ) = 80 + 300 = 380 Since 380 < 400 , this option does not satisfy the first inequality.

Evaluating Option 4 Option 4: 50 minutes walking, 50 minutes basketball w = 50 , b = 50 4 ( 50 ) + 5 ( 50 ) = 200 + 250 = 450 Since 400 ≤ 450 ≤ 600 , this option satisfies both inequalities.

Evaluating Option 5 Option 5: 60 minutes walking, 80 minutes basketball w = 60 , b = 80 4 ( 60 ) + 5 ( 80 ) = 240 + 400 = 640 Since 600"> 640 > 600 , this option does not satisfy the second inequality.

Evaluating Option 6 Option 6: 70 minutes walking, 60 minutes basketball w = 70 , b = 60 4 ( 70 ) + 5 ( 60 ) = 280 + 300 = 580 Since 400 ≤ 580 ≤ 600 , this option satisfies both inequalities.

Final Answer Therefore, the possible solutions are:

50 minutes walking, 50 minutes basketball 70 minutes walking, 60 minutes basketball
Examples
This problem demonstrates how to model real-life constraints using inequalities. For example, a delivery company might want to optimize the number of trucks and drivers they use each day. They have a minimum number of deliveries they need to make (like the minimum calorie burn) and a maximum budget (like the maximum calorie burn). By setting up inequalities, they can find the possible combinations of trucks and drivers that meet their requirements.

Anonymous · Answer

The acceptable options where Joshua meets his caloric burn goals are 50 minutes walking and 50 minutes basketball (Option D) and 70 minutes walking and 60 minutes basketball (Option F). 
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Answer (2)

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