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 0.98 mi. (round to the nearest hundredth as needed.)
(b) what is the solution set in interval notation, when the problem is solved analytically?
a. (varnothing)
b. (left(-infty,\text{type an integer or a simplified fraction.}
ight))
c. (left(\text{type an integer or a simplified fraction.}
ight))
d. ((-infty,infty))

Explanation:

Step1: Set up the inequality

We know that clouds do not form when 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: $68>55 - 5.8x+19x$. Then simplify the right - hand side: $68>55 + 13.2x$. Next, 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:

The solution set in interval notation is $(-\infty,0.98)$

So the answer is that the solution set in interval notation is $(-\infty,0.98)$ which corresponds to an open - interval from negative infinity to 0.98. This means that clouds will not form at altitudes $x$ such that $x < 0.98$ miles.