Barbara drives between Miami, Florida and West Palm Beach, Florida. she drives 45 miles in clear weather and then encounters a thunderstorm for the last 57 miles. she drives 22 mph slower in the thunderstorm than she does in clear whether. If the total time for the trip is 2.25 hours determiner her speed driving in nice weather and her Speed driving in the thunderstorm

Let [tex]v_1[/tex] be the speed and [tex]t_1[/tex] be the time Barbara drives in clear weather, and let [tex]v_2[/tex] be the speed and [tex]t_2[/tex] be the time she drives in the thunderstorm.

Barbara drives 22 mph lower in the thunderstorm than in the clear weather; therefore,

(1). [tex]v_2 = v_1 -22[/tex]

Also,

(2). [tex]v_1t_1 = 45miles[/tex]

(3). [tex]v_2 t_2 = 57[/tex],

and

(4). [tex]t_1+t_2 = 2.25hr[/tex]

From equations (2) and (3) we get:

[tex]t_1 = \dfrac{45}{v_1}[/tex]

[tex]t_2 =\dfrac{57}{v_2}[/tex]

putting these in equation (4) we get:

[tex]\dfrac{45}{v_1}+\dfrac{57}{v_2}=2.25[/tex]

and substituting for [tex]v_2[/tex] from equation (1) we get:

[tex]\dfrac{45}{v_1}+\dfrac{57}{v_1-22}=2.25[/tex]

This equation can be rewritten as

[tex]2.25v_1^2-151.5v_1+990=0[/tex]

which has solutions

[tex]v_1 = 60[/tex]

[tex]v_1 = 7.33[/tex]

We take the first solution [tex]v_1 =60[/tex] because it gives a positive value for [tex]v_2:[/tex]

[tex]v_2 = v_1 -22[/tex]

[tex]v_2 = 60 -22\\[/tex]

[tex]v_2 = 38[/tex].

Thus, Barbara's speed in clear weather is [tex]60mph[/tex] and in the thunderstorm is [tex]38mph[/tex].