Find a model for the Body Mass Index (BMI) of a person, given that BMI varies directly as a person's weight in pounds and inversely as the square of the person's height in inches. If a $7 \mathrm{ft}$ tall person weighing $192 \mathrm{lb}$ has a BMI of 19.13 , how much weight would a $5 \mathrm{ft}$, 7 in. tall person weighing $128 \mathrm{lb}$ need to gain or lose to have a BMI of 21 ? Write your answer as a positive value for weight gain or a negative value for weight loss. Round your answer to one decimal place.
Answer

Step 1 ：Given that a 7ft tall person weighing 192lb has a BMI of 19.13, we can use this information to find the constant of variation (k) in the formula: \(BMI = k \cdot \frac{weight}{height^2}\)

Step 2 ：Convert the height from feet to inches: \(height1 = 7 \times 12 = 84\) inches

Step 3 ：Substitute the given values into the formula to find k: \(19.13 = k \cdot \frac{192}{84^2}\), which gives \(k \approx 703.0275\)

Step 4 ：We are asked to find the weight that a 5ft 7in tall person would need to have in order to have a BMI of 21. Convert the height from feet and inches to inches: \(height2 = 5 \times 12 + 7 = 67\) inches

Step 5 ：Substitute the values into the formula to find the weight: \(21 = 703.0275 \cdot \frac{weight2}{67^2}\), which gives \(weight2 \approx 134.0900604883877\) lb

Step 6 ：Subtract the person's current weight of 128lb to find out how much weight they would need to gain or lose: \(weight_diff = 134.0900604883877 - 128 = 6.0900604883877065\) lb

Step 7 ：Round the weight difference to one decimal place: \(weight_diff \approx 6.1\) lb

Step 8 ：Final Answer: The 5ft 7in tall person would need to gain approximately \(\boxed{6.1}\) pounds to have a BMI of 21.