Formula A: `BMI=w/h^2`
Formula B: `BMI=(4w-100)/5`
The formulas above are used in nutrition to estimate the body mass index BMI, in kilograms per square meter, of adults whose weight w ranges between 50 and 100 kilograms and whose height h is measured in meters.
Based on formula B, what is w in terms of BMI?
IF both formulas A and B give the same estimate for BMI, which of the following expressions is equivalent to 4w-100?

Options

A. `w/h^2`
B. `(5w)/h^2`
C. `(5w)/(4h^2)`
D. `(5w+100)/(4h^2)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Set the two formulas equal to each other
Since both formulas A and B give the same estimate for BMI, we can set them equal to each other:
\[ \frac{w}{h^2} = \frac{4w - 100}{5} \]

### Step 2: Cross-multiply to eliminate the fraction
To eliminate the fractions, we can cross-multiply:
\[ 5w = (4w - 100)h^2 \]

### Step 3: Distribute on the right side
Now, we distribute \(h^2\) on the right side:
\[ 5w = 4wh^2 - 100h^2 \]

### Step 4: Rearrange the equation
Next, we will rearrange the equation to isolate terms involving \(w\):
\[ 5w - 4wh^2 = -100h^2 \]

### Step 5: Factor out \(w\) from the left side
Now, we can factor \(w\) out from the left side:
\[ w(5 - 4h^2) = -100h^2 \]

### Step 6: Solve for \(w\)
Now, we can solve for \(w\):
\[ w = \frac{-100h^2}{5 - 4h^2} \]

### Step 7: Substitute \(w\) back into the equation for \(4w - 100\)
Now, we need to find \(4w - 100\):
\[ 4w - 100 = 4\left(\frac{-100h^2}{5 - 4h^2}\right) - 100 \]

### Step 8: Simplify the expression
Now, we simplify:
\[ 4w - 100 = \frac{-400h^2}{5 - 4h^2} - 100 \]
To combine the terms, we need a common denominator:
\[ = \frac{-400h^2 - 100(5 - 4h^2)}{5 - 4h^2} \]
\[ = \frac{-400h^2 - 500 + 400h^2}{5 - 4h^2} \]
\[ = \frac{-500}{5 - 4h^2} \]

### Final Expression
Thus, the expression equivalent to \(4w - 100\) is:
\[ 4w - 100 = \frac{-500}{5 - 4h^2} \]

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