when the air temperature reaches the dew point, fog may form. this phenomenon also causes clouds to form at…

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? oa. 0 ob. (infty) (type an integer or a simplified fraction.) oc. 0 (type an integer or a simplified fraction.) od. ((-infty,infty))

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? oa. 0 ob. (infty) (type an integer or a simplified fraction.) oc. 0 (type an integer or a simplified fraction.) od. ((-infty,infty))

Answer

Explanation:

Step1: Set up the inequality

Clouds will not form when $T(x)>D(x)$. Given $T_0 = 68$, $D_0=55$, we have $T(x)=68 - 19x$ and $D(x)=55 - 5.8x$. So the inequality is $68 - 19x>55 - 5.8x$.

Step2: Rearrange the inequality

Add $19x$ to both sides: $68>55 - 5.8x+19x$. Then $68>55 + 13.2x$.

Step3: Solve for $x$

Subtract 55 from both sides: $68 - 55>13.2x$, so $13>13.2x$. Divide both sides by 13.2: $x<\frac{13}{13.2}\approx0.98$. In interval - notation, the solution set is $[0, \frac{13}{13.2})$.

Answer:

C. $[0,\frac{13}{13.2})$