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, $w$, and 5 calories per minute spent playing basketball, $b$, 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
Answer (2)
Substitute each pair of walking and basketball minutes into the inequalities.
Check if the resulting value falls between 400 and 600 calories.
Identify the pairs that satisfy both inequalities.
The possible solutions are (50, 50) and (70, 60).
Explanation
Understanding the Problem We are given two inequalities that model the number of calories Joshua burns per day by walking ( w ) and playing basketball ( b ): = 400"> 4 w + 5 b " >= 400 and 4 w + 5 b " <= 600 . We need to check which of the given pairs ( w , b ) satisfy both inequalities.
Testing Each Pair Let's test each pair:
(40 minutes walking, 40 minutes basketball): 4 ( 40 ) + 5 ( 40 ) = 160 + 200 = 360 . Since 360 < 400 , this pair does not satisfy the first inequality, so it's not a solution.
(60 minutes walking, 20 minutes basketball): 4 ( 60 ) + 5 ( 20 ) = 240 + 100 = 340 . Since 340 < 400 , this pair does not satisfy the first inequality, so it's not a solution.
(20 minutes walking, 60 minutes basketball): 4 ( 20 ) + 5 ( 60 ) = 80 + 300 = 380 . Since 380 < 400 , this pair does not satisfy the first inequality, so it's not a solution.
(50 minutes walking, 50 minutes basketball): 4 ( 50 ) + 5 ( 50 ) = 200 + 250 = 450 . Since 400" <= 450" <= 600 , this pair is a solution.
(60 minutes walking, 80 minutes basketball): 4 ( 60 ) + 5 ( 80 ) = 240 + 400 = 640 . Since 600"> 640 > 600 , this pair does not satisfy the second inequality, so it's not a solution.
(70 minutes walking, 60 minutes basketball): 4 ( 70 ) + 5 ( 60 ) = 280 + 300 = 580 . Since 400" <= 580" <= 600 , this pair is a solution.
Finding the Solutions Therefore, the possible solutions are:
50 minutes walking, 50 minutes basketball
70 minutes walking, 60 minutes basketball
Examples
Understanding inequalities can help manage fitness goals. For example, if you want to maintain a certain calorie range through exercise, you can use inequalities to model the relationship between different activities and their calorie burn rates. By setting up inequalities based on your desired calorie range and the calories burned per minute for each activity, you can determine the possible combinations of exercise times that meet your goals. This approach allows for a flexible and personalized fitness plan.
The possible pairs that satisfy Joshua's calorie burn requirement are 50 minutes of walking and 50 minutes of basketball, and 70 minutes of walking and 60 minutes of basketball. Only these combinations result in burning between 400 and 600 calories. Other options either burn too few or too many calories.
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