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For the three-part question that follows, provide your answer to each part in the given workspace. Identify each part with a coordinating response. Be sure to clearly label each part of your response as Part A, Part B, and Part C.
Mike is trying to lose weight to enter a different weight class for his wrestling team next year. After eight weeks, he weighed 255lbs. After 12 weeks, he weighed 245lbs.
Part A: Write an equation to model the relationship. Show all work.
Part B: Explain what the equation from Part A represents to the problem.
Part C: Mike's goal weight is 180 lbs. How long will it take him to reach his goal? Show all work.
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Part A:
Let's denote the weeks as "w" and the weight in pounds as "p". We can assume that the weight loss is linear over time (weeks). To find the equation that models the relationship, we can use the slope-intercept form of a linear equation:
p = mx + b
where "m" represents the slope and "b" represents the y-intercept. We can determine the slope by finding the change in weight divided by the change in weeks:
Slope (m) = (245lbs - 255lbs) / (12 weeks - 8 weeks) = -10lbs / 4 weeks
Simplifying, we have:
m = -10/4 = -2.5
Since we know the weight and corresponding weeks at a specific point (8 weeks), we can substitute those values into the equation to solve for the y-intercept (b). Let's use the given weight of 255lbs and 8 weeks:
255lbs = -2.5 * 8 weeks + b
Now, let's solve for b:
255lbs = -20lbs + b
b = 275lbs
Therefore, the equation that models the relationship between weight and weeks is:
p = -2.5w + 275
Part B:
The equation from Part A represents the relationship between Mike's weight (p) and the number of weeks (w). It shows that Mike's weight decreases linearly over time at a rate of 2.5 pounds per week. The positive y-intercept of 275 pounds indicates his starting weight at week 0.
Part C:
To find out how long it will take Mike to reach his goal weight of 180 pounds, we can substitute the goal weight into the equation and solve for the number of weeks (w):
180lbs = -2.5w + 275
Let's solve for w:
-2.5w = 180lbs - 275
-2.5w = -95lbs
w = -95lbs / -2.5
w ≈ 38
So, it will take Mike approximately 38 weeks to reach his goal weight of 180 pounds.