If for the first 150 miles of a trip, an car drives at v mph, for the next 200 miles, the car drives at (v+25) mph and the average speed of the whole trip is 35 mph, then the value of v will be 20mph (A).
Given Information:
Average speed = 35 mph
Total distance = 150 + 200 = 350 miles
For the first 150 miles of a trip, an car drives at v mph speed
⇒ Time, t1 = 150/v hrs
For the next 200 miles, the car drives at (v+25) mph speed
⇒ Time, t1 = 200 / (v+25) hrs
Now, the formula for average speed is given as,
Total distance / Total time
= [tex]\frac{350}{\frac{150}{v}+\frac{200}{(v+25)} }[/tex]
⇒ [tex]\frac{350}{\frac{150}{v}+\frac{200}{(v+25)} }[/tex] = 35 mph
[tex]\frac{350 v (v+25)}{150(v+25) + 200v}[/tex] = 35
[tex]\frac{350v^{2} + 8750v}{150v +3750 + 200v}[/tex] = 35
350v² + 8750v = 35 (350v + 3750)
v² + 25v = 35v + 375
v² - 10v = 375
v² - 10v - 375 = 0
Now, the above equation for speed can be written as,
v² - 20v + 15v - 375 = 0
v(v-20) + 15(v-20) = 0
(v-20) (v+15) = 0
v = 20
or v = -15
Since, speed is a scalar quantity, it can't be negative. Thus, v = 20 mph
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