To derive a formula for the boiling point of water \( b \) in degrees Fahrenheit at an elevation \( h \) in feet, we note the following information:
- At sea level (0 feet), the boiling point is \( 212°F \).
- The boiling point decreases by \( 1.72°F \) for every \( 1,000 \) feet of elevation.
Using this information, we can formulate the relationship:
- For every \( 1,000 \) feet of elevation, the boiling point decreases by \( 1.72°F \). Therefore, for every foot of elevation, the decrease is: \[ \frac{1.72°F}{1000 \text{ feet}} = 0.00172°F \text{ per foot} \]
From this, we can write the formula for boiling point \( b \) in terms of elevation \( h \): \[ b = 212°F - 0.00172 \cdot h \]
Next, we need to solve the formula for the elevation \( h \) when the boiling point for water is \( 190°F \):
- Set \( b = 190°F \): \[ 190 = 212 - 0.00172 \cdot h \]
- Rearranging the equation to solve for \( h \): \[ 0.00172 \cdot h = 212 - 190 \] \[ 0.00172 \cdot h = 22 \]
- Now, divide both sides by \( 0.00172 \) to find \( h \): \[ h = \frac{22}{0.00172} \]
- Calculating this gives: \[ h \approx 12790.70 \text{ feet} \]
Thus, the formula for the boiling point \( b \) in degrees Fahrenheit at elevation \( h \) in feet is: \[ b = 212 - 0.00172h \] And the elevation \( h \) when the boiling point is \( 190°F \) is approximately \( 12,791 \) feet.