Answer :
To determine how many weeks a wrestler should lose weight to qualify in his weight class, we need to find the number of weeks it takes for his weight to be more than 165 pounds and less than or equal to 185 pounds. Here’s how we can solve this step-by-step:
1. Current Weight and Rate of Loss:
- The wrestler currently weighs 189 pounds.
- He is losing weight at a rate of 0.5 pounds per week.
2. Qualifying Weight Range:
- Minimum weight needed for qualifying: more than 165 pounds.
- Maximum weight allowed for qualifying: less than or equal to 185 pounds.
3. Setting Up the Inequality:
To be in the qualifying range, we need:
[tex]\(165 < 189 - 0.5w \leq 185\)[/tex]
4. Solving the Inequalities:
- For the upper limit: [tex]\(189 - 0.5w \leq 185\)[/tex]
- Subtract 189 from both sides: [tex]\(-0.5w \leq -4\)[/tex]
- Divide by [tex]\(-0.5\)[/tex] (remember to flip the inequality sign):
[tex]\(w \geq 8\)[/tex]
- For the lower limit: [tex]\(165 < 189 - 0.5w\)[/tex]
- Subtract 189 from both sides: [tex]\(-24 < -0.5w\)[/tex]
- Divide by [tex]\(-0.5\)[/tex] (flip the inequality sign):
[tex]\(w > 48\)[/tex]
5. Conclusion:
Combining both results, the wrestler should lose weight for more than 48 weeks to ensure his weight is more than 165 pounds. However, based on the upper limit, he would meet the maximum qualifying weight after about 8 weeks.
Given these findings, technically, it’s not possible for the wrestler to be within the qualifying range if he continues losing at this rate, because the maximum weeks that allow him to be under 185 pounds (8 weeks) is less than the minimum weeks needed to go under 165 pounds (48 weeks).
1. Current Weight and Rate of Loss:
- The wrestler currently weighs 189 pounds.
- He is losing weight at a rate of 0.5 pounds per week.
2. Qualifying Weight Range:
- Minimum weight needed for qualifying: more than 165 pounds.
- Maximum weight allowed for qualifying: less than or equal to 185 pounds.
3. Setting Up the Inequality:
To be in the qualifying range, we need:
[tex]\(165 < 189 - 0.5w \leq 185\)[/tex]
4. Solving the Inequalities:
- For the upper limit: [tex]\(189 - 0.5w \leq 185\)[/tex]
- Subtract 189 from both sides: [tex]\(-0.5w \leq -4\)[/tex]
- Divide by [tex]\(-0.5\)[/tex] (remember to flip the inequality sign):
[tex]\(w \geq 8\)[/tex]
- For the lower limit: [tex]\(165 < 189 - 0.5w\)[/tex]
- Subtract 189 from both sides: [tex]\(-24 < -0.5w\)[/tex]
- Divide by [tex]\(-0.5\)[/tex] (flip the inequality sign):
[tex]\(w > 48\)[/tex]
5. Conclusion:
Combining both results, the wrestler should lose weight for more than 48 weeks to ensure his weight is more than 165 pounds. However, based on the upper limit, he would meet the maximum qualifying weight after about 8 weeks.
Given these findings, technically, it’s not possible for the wrestler to be within the qualifying range if he continues losing at this rate, because the maximum weeks that allow him to be under 185 pounds (8 weeks) is less than the minimum weeks needed to go under 165 pounds (48 weeks).
