f the air temperature at ground level is 90degrees°F, the air temperature x miles high is given by T(x)equals=90minus−19x. Determine the altitudes at which the air temperature is from 52degrees°F to 42.5degrees°F.
To determine the altitudes at which the air temperature is between 52°F and 42.5°F, we need to solve the inequality T(x) ≤ 52 and T(x) ≥ 42.5.
Given that T(x) = 90 - 19x, we can substitute T(x) into the inequality:
90 - 19x ≤ 52
and
90 - 19x ≥ 42.5
Let's solve each inequality separately:
1) 90 - 19x ≤ 52:
First, subtract 90 from both sides to isolate the variable:
-19x ≤ 52 - 90
Simplifying the right side:
-19x ≤ -38
Now, divide both sides by -19. Since we are dividing by a negative number, the inequality sign will change direction:
x ≥ (-38) / (-19)
x ≥ 2
So, the altitude at which the air temperature is at or below 52°F is at 2 miles high.
2) 90 - 19x ≥ 42.5:
First, subtract 90 from both sides to isolate the variable:
-19x ≥ 42.5 - 90
Simplifying the right side:
-19x ≥ -47.5
Now, divide both sides by -19. Since we are dividing by a negative number, the inequality sign will change direction:
x ≤ (-47.5) / (-19)
x ≤ 2.5
So, the altitude at which the air temperature is at or above 42.5°F is at 2.5 miles high.
Therefore, the altitudes at which the air temperature is between 52°F and 42.5°F are between 2 miles and 2.5 miles high.
To determine the altitudes at which the air temperature is between 52°F and 42.5°F, we need to solve the inequality T(x) < 52 and T(x) > 42.5.
Given that T(x) = 90 - 19x, we can rewrite the inequality as:
52 > 90 - 19x (equation 1)
42.5 < 90 - 19x (equation 2)
Let's solve equation 1 first:
52 > 90 - 19x
-38 > -19x
2 < x
Now, let's solve equation 2:
42.5 < 90 - 19x
-47.5 < -19x
2.5 > x
So, the solution will be 2.5 < x < 2.
This means that the air temperature is between 52°F and 42.5°F at altitudes between 2.5 and 2 miles.
42.5 <= 90-19x <= 52
-47.5 <= -19x <= -38
2.5 >= x >= 2