Answer :
To determine how many weeks the wrestler should lose weight to qualify for his weight class, we need to understand the problem and the conditions.
### Conditions for the Weight Class:
- The wrestler's weight needs to be more than 165 pounds.
- The wrestler's weight should be less than or equal to 185 pounds.
### Current Situation:
- The wrestler currently weighs 189 pounds.
- He is losing 0.5 pounds per week.
We need to calculate the number of weeks it will take for the wrestler to weigh more than 165 pounds but less than or equal to 185 pounds and choose the correct model expression describing this situation.
### Steps:
1. Calculate Weeks to Reach Upper Weight Limit:
- He needs his weight to be 185 pounds or less.
- Current weight = 189 pounds.
- Weight loss goal to reach 185 pounds:
[tex]\( 189 - 185 = 4 \)[/tex] pounds.
- At a rate of 0.5 pounds per week, weeks to reach 185 pounds:
[tex]\( \frac{4}{0.5} = 8 \)[/tex] weeks.
2. Calculate Weeks to Reach Lower Weight Limit (by being less):
- He needs to be more than 165 pounds.
- Current weight = 189 pounds.
- Weight loss to lose and still be more than 165 pounds:
[tex]\( 189 - 165 = 24 \)[/tex] pounds.
- At a rate of 0.5 pounds per week, weeks to reach just above 165 pounds:
[tex]\( \frac{24}{0.5} = 48 \)[/tex] weeks.
### Choose the Correct Model:
The correct expression that models the required weight range as the wrestler loses weight over weeks is:
[tex]\[ 165 \leq 189 - 0.5w < 185 \][/tex]
This expression ensures:
- Weight stays more than 165 (but not equal to),
- Weight goes below (but not equal to) 185.
So, the wrestler should aim for week 9 specifically for meeting both conditions (not just touching either boundary but staying within and satisfying both), but he reaches the specific boundaries by week 8 and week 48. However, the core operational window or proof of qualification starts with crossing that threshold around week 8 (upper boundary within limits).
This is how we determine the time frame and correct coverage with the inequality model.
### Conditions for the Weight Class:
- The wrestler's weight needs to be more than 165 pounds.
- The wrestler's weight should be less than or equal to 185 pounds.
### Current Situation:
- The wrestler currently weighs 189 pounds.
- He is losing 0.5 pounds per week.
We need to calculate the number of weeks it will take for the wrestler to weigh more than 165 pounds but less than or equal to 185 pounds and choose the correct model expression describing this situation.
### Steps:
1. Calculate Weeks to Reach Upper Weight Limit:
- He needs his weight to be 185 pounds or less.
- Current weight = 189 pounds.
- Weight loss goal to reach 185 pounds:
[tex]\( 189 - 185 = 4 \)[/tex] pounds.
- At a rate of 0.5 pounds per week, weeks to reach 185 pounds:
[tex]\( \frac{4}{0.5} = 8 \)[/tex] weeks.
2. Calculate Weeks to Reach Lower Weight Limit (by being less):
- He needs to be more than 165 pounds.
- Current weight = 189 pounds.
- Weight loss to lose and still be more than 165 pounds:
[tex]\( 189 - 165 = 24 \)[/tex] pounds.
- At a rate of 0.5 pounds per week, weeks to reach just above 165 pounds:
[tex]\( \frac{24}{0.5} = 48 \)[/tex] weeks.
### Choose the Correct Model:
The correct expression that models the required weight range as the wrestler loses weight over weeks is:
[tex]\[ 165 \leq 189 - 0.5w < 185 \][/tex]
This expression ensures:
- Weight stays more than 165 (but not equal to),
- Weight goes below (but not equal to) 185.
So, the wrestler should aim for week 9 specifically for meeting both conditions (not just touching either boundary but staying within and satisfying both), but he reaches the specific boundaries by week 8 and week 48. However, the core operational window or proof of qualification starts with crossing that threshold around week 8 (upper boundary within limits).
This is how we determine the time frame and correct coverage with the inequality model.
