Question

It is 76f at the 6000-foot level of a mountain, and 49f 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 thousands in feet.

Answer 1

The temperature (t) at a given elevation (e) on a mountain in thousands of feet can be found with the linear equation t = -4.5e + 103, where the slope is -4.5°F per thousand feet, and the y-intercept is 103°F.

Step-by-step explanation:

To find the temperature t at an elevation e on the mountain in thousands of feet, we first need to determine the slope of the temperature change rate between two known points. Given the temperature at a 6000-foot level is 76°F and at a 12000-foot level is 49°F, we can calculate the slope (m) as the ratio of the change in temperature to the change in elevation. We know the temperatures at two different elevations, so:

• Temperature at 6000 feet: t1 = 76°F
• Temperature at 12000 feet: t2 = 49°F
• Elevation at point 1: e1 = 6
• Elevation at point 2: e2 = 12

Using these points, we can calculate the slope (m) of the temperature:

m = (t2 - t1) / (e2 - e1)

m = (49 - 76) / (12 - 6)

m = -27 / 6

m = -4.5°F per thousand feet

Next, we calculate the y-intercept (b) using one of the known points. For convenience, we'll use the point at 6000 feet:

b = t1 - m * e1

b = 76 - (-4.5) * 6

b = 76 + 27

b = 103°F

Now, we can write the linear equation in slope-intercept form (y = mx + b), where t is the temperature and e is the elevation in thousands of feet: t = -4.5e + 103.