By analyzing the temperature-altitude relationship on a mountain, we can establish a linear model that describes how temperature changes with varying altitudes.
Explanation:
a) Knowing the altitude and temperature data, we can use the formula for the slope of a line:
m = (y2 - y1) / (x2 - x1),
where (x1, y1) and (x2, y2) are two points on the line.
Using the altitude and temperature values provided:
(x1, y1) = (4800, 79) and (x2, y2) = (6400, 67),
we can calculate the slope:
m = (67 - 79) / (6400 - 4800)
= -12 / 1600
= -0.0075.
Now, to find the y-intercept (b), we can substitute one of the points (4800, 79) into the equation:
79 = (-0.0075)(4800) + b.
Solving for b, we have:
b = 79 + 0.0075(4800)
= 79 + 36
= 115.
Therefore, the linear model is T(x) = -0.0075x + 115.
b) The units of the slope (m) can be determined by looking at the units of the y values (temperature) and x values (altitude). In this case, the units of temperature are degrees, and the units of altitude are feet. Therefore, the units of the slope (m) are degrees per foot.
c) To find T(5800), we can substitute x = 5800 into the linear model:
T(5800) = -0.0075(5800) + 115
= -43.5 + 115
= 71.5.
Answers:
a) T(x) = -0.0075x + 115.
b) The units of the slope (m) can be determined by looking at the units of the y values (temperature) and x values (altitude). In this case, the units of temperature are degrees, and the units of altitude are feet. Therefore, the units of the slope (m) are degrees per foot.
c) 71.5 degrees.