Final answer:
The linear equation representing the temperature T at an elevation x (in thousands of feet) on a mountain is T = -4.5x + 103, where the slope is -4.5, indicating a decrease of 4.5 degrees Fahrenheit for every thousand feet of elevation.
Step-by-step explanation:
The problem is asking us to find a linear equation that represents the temperature at various altitudes on a mountain. To solve this, let's use the two points provided: (6, 76) and (12, 49), where the first coordinate represents the altitude in thousands of feet and the second coordinate represents the temperature in degrees Fahrenheit. We need to calculate the slope (m) of the line that passes through these points.
The slope (m) is calculated as the change in temperature divided by the change in altitude (in thousands of feet).
m = (Temperature change) / (Altitude change) = (49 - 76) / (12 - 6) = (-27) / (6) = -4.5
Now we have the slope, which is -4.5°F per thousand feet. Next, we use point-slope form to find our linear equation. Let's use the point (6, 76) and our slope:
T - 76 = -4.5(x - 6)
To write this in slope-intercept form, we solve for T:
T = -4.5x + 27 + 76
T = -4.5x + 103
So the linear equation representing the temperature T at an elevation of x thousand feet is T = -4.5x + 103.