In certain conditions, the air temperature drops about 3.1F for each 1,000-foot rise in altitude.
(A) If the temperature at sea level is 74 F, write a linear equation that expresses the temperature T in terms of altitude A in thousands of feet.
(B) At what altitude is the temperature 43F?
(A) To express the temperature T in terms of altitude A in thousands of feet, we can use the slope-intercept form of a linear equation:
T = mx + b
Given that the air temperature drops about 3.1F for each 1,000-foot rise in altitude, the slope (m) of the equation will be -3.1. Since the temperature at sea level is 74 F, we can substitute this value into the equation and solve for the y-intercept (b):
74 = -3.1(0) + b
74 = b
Therefore, the linear equation that expresses the temperature T in terms of altitude A in thousands of feet is:
T = -3.1A + 74
(B) To find the altitude at which the temperature is 43 F, we can set T = 43 in the equation:
43 = -3.1A + 74
Rearranging the equation, we get:
-3.1A = 43 - 74
-3.1A = -31
Dividing both sides by -3.1, we find:
A = -31 / -3.1
A ≈ 10
Therefore, the altitude at which the temperature is 43 F is approximately 10,000 feet.
(A) To write a linear equation that expresses the temperature T in terms of altitude A, we first need to determine the rate of change. Since the air temperature drops 3.1°F for each 1,000-foot rise in altitude, the rate of change is -3.1°F per 1,000 feet.
Next, we need to find the initial temperature at sea level, which is 74°F. This will be our y-intercept.
So, the linear equation that expresses the temperature T in terms of altitude A is:
T = -3.1A + 74
(B) To find the altitude at which the temperature is 43°F, we need to solve the equation:
43 = -3.1A + 74
Let's solve for A:
-3.1A + 74 = 43
-3.1A = 43 - 74
-3.1A = -31
A = -31 / -3.1
A ≈ 10
Therefore, the altitude at which the temperature is 43°F is approximately 10,000 feet.
To write a linear equation relating temperature to altitude, we can use the point-slope form of a linear equation, which is given by:
y - y1 = m(x - x1)
Here, 'y' represents the temperature (T), 'x' represents the altitude (A) in thousands of feet, 'm' represents the rate at which the temperature decreases with altitude (which is -3.1°F per 1,000-foot rise), and (x1, y1) represents any point on the line.
(A) Let's choose the temperature at sea level (0 altitude) as our point (x1, y1) = (0, 74°F). Plugging these values into the point-slope form, we have:
T - 74 = -3.1(A - 0)
Simplifying further:
T - 74 = -3.1A
Now we can rewrite it in terms of T:
T = -3.1A + 74
So, the linear equation that expresses the temperature (T) in terms of altitude (A) in thousands of feet is T = -3.1A + 74.
(B) To find the altitude at which the temperature is 43°F, we substitute T = 43 into the equation and solve for A:
43 = -3.1A + 74
Subtracting 74 from both sides:
-31 = -3.1A
Dividing both sides by -3.1:
A ≈ 10
Therefore, at an altitude of approximately 10,000 feet, the temperature would be 43°F.