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 $68^{circ}f$ and the dew - point is $55^{circ}f$. 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. (a) the altitudes at which clouds will not form is below $square$ mi. (round to the nearest hundredth as needed.)

Explanation:

Step1: Set up the inequality

We know that clouds will not form when the air - temperature $T(x)$ is above the dew - point $D(x)$. Given $T_0 = 68$ and $D_0=55$, $T(x)=68 - 19x$ and $D(x)=55 - 5.8x$. So we set up the inequality $68 - 19x>55 - 5.8x$.

Step2: Solve the inequality for $x$

First, add $19x$ to both sides of the inequality:
$68>55 - 5.8x+19x$.
Simplify the right - hand side: $68>55 + 13.2x$.
Then subtract 55 from both sides: $68 - 55>13.2x$, which gives $13>13.2x$.
Finally, divide both sides by 13.2: $x<\frac{13}{13.2}\approx0.98$.

Answer:

$0.98$