Question

It is 76º F at the 6000-foot level of a mountain, and 49º F at the 12,000-foot level of the mountain. Write a linear equation, in slope-intercept form, to find the temperature T at an elevation e on the mountain, where e is in thousands of feet.

Answer 1

The linear equation to determine temperature at a given elevation on the mountain is T(e) = 76 - 4.5(e - 6), where T is the temperature in Fahrenheit and e is the elevation in thousands of feet.

Step-by-step explanation:

The goal is to write a linear equation to determine the temperature T at any given elevation e (in thousands of feet) on a mountain. To find the slope, we use the rate of change between two points: (6000 feet, 76°F) and (12000 feet, 49°F). The slope m is the ratio of the change in temperature to the change in elevation (in thousands of feet).

First, calculate the slope:
m = (Temperature Change) / (Elevation Change) = (49°F - 76°F) / (12000 feet - 6000 feet) = -27°F / 6000 feet = -0.0045°F/foot.

Using e for elevation (in thousands of feet) and the point-slope form of the line, which passes through the point (6, 76), the equation is:
T(e) = 76 + (-0.0045°F/foot) × (e - 6000 feet).

Since we want the elevation in thousands of feet, let elevation e be in thousands. So, if e = 6 corresponds to 6000feet, then:

T(e) = 76 + (-0.0045°F/foot) × (e - 6)

Finally, convert the slope to a per thousand feet basis and you get:

T(e) = 76 - 4.5(e - 6)

Thus, the slope-intercept form of the temperature equation on the mountain as a function of elevation in thousands of feet is:

T(e) = 76 - 4.5(e - 6)