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Air temperature decreases as altitude increases. If the ground temperature is 62 degrees F, then the air temperature \( T \) at \( x \) miles high is given by \( T = 62 - 19x \).

(a) Determine the altitudes \( x \) where the temperature \( T \) is between 6 degrees F and 34 degrees F.

(b) Use an absolute value inequality to describe these altitudes.

Answer :

Answer:

(a) 1.47 ≤ x ≤ 2.95

(b) | x − 2.21 | ≤ 0.74

Step-by-step explanation:

Given the temperature as a function of altitude, we can substitute in values for temperature and solve for the corresponding altitudes. To convert this interval to an absolute value inequality, we must first find the average and the range. The average of two numbers is half their sum, and the range of two numbers is the difference between them.

(a) We are given the equation T = 62 − 19x, where T is temperature in Fahrenheit and x is altitude in miles. When the temperature is 6F, the altitude is:

T = 62 − 19x

6 = 62 − 19x

19x = 56

x = 2.95

When the temperature is 34F, the altitude is:

T = 62 − 19x

34 = 62 − 19x

19x = 28

x = 1.47

The interval of altitudes x where the temperature is between 6F and 34F are therefore:

1.47 ≤ x ≤ 2.95

(b) We want to convert this interval to an absolute value inequality. The interval a ≤ x ≤ b can be converted to the following:

[tex]\Large \text {$ |x-\mu|\leq\ $} \huge \text {$ \frac{R}{2} $}[/tex]

where μ is the average of a and b and R is the range or difference between a and b. In other words, the inequality is:

[tex]\Large \text {$ |x-$} \huge \text {$ \frac{a+b}{2} $} \Large \text {$ |\leq\ $} \huge \text {$ \frac{b-a}{2} $}[/tex]

In this case, a = 1.47 and b = 2.95. The average is:

[tex]\Large \text {$ \mu=\ $} \huge \text {$ \frac{a+b}{2} $}\\\\\Large \text {$ \mu=\ $} \huge \text {$ \frac{1.47+2.95}{2} $}\\\\\Large \text {$ \mu=2.21 $}[/tex]

And half of the range is:

[tex]\huge \text {$ \frac{R}{2} $} \Large \text {$ \ =\ $} \huge \text {$ \frac{b-a}{2} $}\\\\\huge \text {$ \frac{R}{2} $} \Large \text {$ \ =\ $} \huge \text {$ \frac{2.95-1.47}{2} $}\\\\\huge \text {$ \frac{R}{2} $} \Large \text {$ \ =0.74 $}[/tex]

Therefore, the absolute value inequality is:

[tex]\Large \text {$ |x-\mu|\leq\ $} \huge \text {$ \frac{R}{2} $}\\\\\Large \text {$ |x-2.21|\leq 0.74 $}[/tex]

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