Question

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 ( t_{0} ) and ( d_{0} ) respectively, the air temperature at an altitude of ( x ) miles can be approximated by ( t(x)=t_{0}-19x ), and the dew point can be approximated by ( d(x)=d_{0}-5.8x ). suppose the ground - level temperature is ( 61^{circ}f ) and the dew point is ( 46^{circ}f ). note that clouds will not form at altitudes below when the air temperature is above the dew point. (a) use the intersection - of - graphs method to estimate the altitude at which clouds will not form. (b) solve part (a) analytically. (a) the altitude at which clouds will not form is below (square) mi (round to the nearest hundredth as needed.)

Explanation:

Step1: Substitute the given values

Given \(T_0 = 61\) and \(D_0=46\), so \(T(x)=61 - 19x\) and \(D(x)=46 - 5x\).

Step2: Set up the inequality

Clouds will not form when \(T(x)>D(x)\). So, \(61 - 19x>46 - 5x\).

Step3: Solve the inequality

First, add \(19x\) to both sides: \(61>46 - 5x+19x\), which simplifies to \(61>46 + 14x\).
Then subtract \(46\) from both sides: \(61 - 46>14x\), so \(15>14x\).
Finally, divide both sides by \(14\): \(x<\frac{15}{14}\approx1.07\).

Answer:

The altitude at which clouds will not form is below \(1.07\) mi.