Question

As mountain climbers know, the higher you go, the cooler the temperature gets. At noon on July 4th last summer, the temperature at the top of Mt. Washington — elevation 6288 feet — was 56◦F. The temperature at base camp in Pinkham Notch — elevation 2041 feet — was 87◦F. It was a clear, still day. At that moment, a group of hikers reached Tuckerman Junction — elevation 5376 feet. To the nearest degree, calculate the temperature the hikers were experiencing at that time and place. When you decided how to model this situation, what assumptions did you make?

Answer 1

The temperature at 5376 ft is approximately 63°F

The assumption made was that the temperature varies linearly with elevation

Explanation:

The parameters given are;

Temperature at 6288 feet = 56°F = 286.5

Temperature at 2041 feet = 87°F = 303.71

We are to find the temperature at 5376 feet

Let the temperature be the y-coordinate value and the elevation be the x-coordinate value, to find the temperature, we have the temperature gradient given by the relation;

`m = (y_2-y_1)/(x_2 - x_1) = (303.71-286.5)/(2041 - 6288)= -4.05 * 10^(-3) \ K/ft`

The temperature at 5376 ft will be the temperature at 2041 added to the decrease in temperature from climbing to 5376 ft

The increase in elevation is 5376 - 2041 = 3335 ft

The decrease in temperature = 3335 ft × (-4.05 × 10⁻³) K/ft = -13 .5 K

The temperature at 5376 ft will then be 303.71 - 13.5 = 290.196 K = 62.68°F ≈ 63°F

The assumption made was that the decrease in temperature with elevation is linear.