Frank Weighs 160 Pounds And Is On A Diet To Gain 2 Pounds A

Frank Weighs 160 Pounds And Is On Diet To Gain 2 Pounds A Week So That

Frank weighs 160 pounds and is on a diet to gain 2 pounds a week so that he can make the football team. John weighs 208 pounds and is on a diet to lose 3 pounds a week so that he can be on the wrestling team in a lower weight class.

a) If Frank and John can meet these goals with their diets, when will they weigh the same, and how much will they weigh at that time?

b) Clearly explain your method. Hint: Use the information in the problem to set up y=mx+b equations for both Frank and John. Define your variables: y = total weight (lbs); x = time (weeks); m = rate of weight change (lbs/week); b = initial weight (lbs). Set the equations and solve for x.

Paper For Above instruction

To determine when Frank and John will weigh the same, and what their common weight will be at that point, we start by modeling their weight changes over time using linear equations based on their initial weights and weekly weight change rates.

Let’s define the variables:

- y: total weight in pounds

- x: time in weeks

- For Frank: initial weight (b) = 160 pounds, weekly weight increase (m) = 2 pounds

- For John: initial weight (b) = 208 pounds, weekly weight decrease (m) = -3 pounds

Using the slope-intercept form y = mx + b:

- Frank's weight after x weeks:

yFrank = 2x + 160

- John's weight after x weeks:

yJohn = -3x + 208

To find the time when Frank and John weigh the same, we set the equations equal:

2x + 160 = -3x + 208

Adding 3x to both sides:

2x + 3x + 160 = 208

5x + 160 = 208

Subtracting 160 from both sides:

5x = 48

Dividing both sides by 5:

x = 9.6 weeks

Now, substitute x back into one of the equations to find the weight at that time:

Using Frank's equation:

y = 2(9.6) + 160 = 19.2 + 160 = 179.2 pounds

Therefore, Frank and John will both weigh approximately 179.2 pounds after about 9.6 weeks.

This indicates that within roughly 9 and a half weeks, Frank will have gained enough weight to match John's decreasing weight, aligning with their respective goals of reaching a common weight for their sports teams.

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