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When the air temperature reaches the dew point, fog may form. This phenomenon also causes clouds to form at higher altitudes. Both
the air temperature and the dew point decrease at a constant rate as the altitude above ground level increases. If the ground-level
temperature and dew point are To and Do, respectively, the air temperature at an altitude of x miles can be approximated by
T(x) = To - 19x, and the dew point can be approximated by D(x) = D.-5.8x. Suppose the ground-level temperature is 65°F and the dew
point is 50°F. Note that clouds will not form at altitudes for which the air temperature is above the dew point.
(a) Use the intersection-of-graphs method to estimate the altitudes at which clouds will not form.
(b) Solve part (a) analytically.
TORRE

Sagot :

Using linear functions, it is found that:

a) Clouds will not form at an altitude below 1.136 miles.

b) The analytical solution to the inequality is x > 1.136.

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The linear function for the temperature at an altitude of x miles is:

[tex]T(x) = T(0) - 19x[/tex]

In which

  • T(0) is the ground level temperature.
  • The ground level temperature is 65ºF, thus [tex]T(0) = 65[/tex], and:

[tex]T(x) = 65 - 19x[/tex]

The function for the dew point is:

[tex]D(x) = D(0) - 5.8x[/tex]

The ground level dew point is 50ºF, thus:

[tex]D(x) = 50 - 5.8x[/tex].

Item a:

  • Clouds will not form if the temperature is above the dew point.
  • In the sketch, temperature is in blue and the dew point is in red.
  • The intersection is at an altitude of 1.136 miles.
  • Before, the temperature is above.
  • Thus, clouds will not form at an altitude below 1.136 miles.

Item b:

Analytically, we have to solve the following inequality:

[tex]T(x) > D(x)[/tex]

[tex]65 - 19x > 50 - 5.8x[/tex]

[tex]-13.2x < -15[/tex]

[tex]13.2x > 15[/tex]

[tex]x > \frac{15}{13.2}[/tex]

[tex]x > 1.136[/tex]

A similar problem is given at https://brainly.com/question/12965245

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