Question

A tourist has planned a trip to cover the distance of 640 miles, driving at some constant speed. However, when he already covered a quarter of the distance, he took a rest for 1.2 hours. Then, in order to arrive at the destination on time, he increased the speed by 20 mph. How long, actually, the trip lasted?

Answer 1

A quarter of the distance = 640 / 4 = 160 miles

rest of the distance = 640 - 160 = 480 miles

Distance, speed, time formula:

s = d / t

If he did not make a rest he would travel:

t = 640 / s1 [ hours ]

where s1 = speed in a first quarter of the distance

t1 = d / s1 = 160 / s1

where t1 =time in a first quarter of the distance

time of a rest = 1.2 [ hours ]

New speed:

s2 = s1 + 20

New time = t2 = rest of the distance / new speed

t2 = 480 / ( s1 + 20 )

In order to arrive at the destination on time mean a trip will last:

640 / s1

how long it would take to travel at a constant speed to lasted s1

Total time:

t = 160 / s1 + 1.2 + 480 / ( s1 + 20 ) = 640 / s1

[ 160 ∙ ( s1 + 20 ) + 1.2 ∙ s1 ∙ ( s1 + 20 ) + 480 ∙ s1 ] / [ s1 ∙ ( s1 + 20 ) ] = 640 / s1

( 160 s1 + 3200 + 1.2 s1² + 24 s1+ 480 s1 ) / [ s1 ∙ ( s1 + 20 ) ] = 640 ∙ ( s1 + 20 ) / [ s1 ∙ ( s1 + 20 ) ]

( 1.2 s1² + 664 s1+ 3200 ) / [ s1 ∙ ( s1 + 20 ) ] = 640 ∙ ( s1 + 20 ) / [ s1 ∙ ( s1 + 20 ) ]

Multiply both sides by s1 ∙ ( s1 + 20 )

1.2 s1² + 664 s1 + 3200 = 640 ∙ ( s1 + 20 )

1.2 s1² + 664 s1 + 3200 = 640 s1 + 12800

Subtract 640 s1 + 12800 to both sides

1.2 s1² + 664 s1 + 3200 - 640 s1 - 12800 = 0

1.2 s1² + 24 s1 - 9600 = 0

Divide both sides by 1.2

s1² + 20 s1 - 8000 = 0

The solutions are:

s1 = - 100

and

s1 = 80

Speed can't be negative so:

s1 = 80 miles / hour

Total time:

t = 160 / s1 + 1.2 + 480 / ( s1 + 20 )

t = 160 / 80 + 1.2 + 480 / ( 80 + 20 )

t = 2 + 1.2 + 480 / 100

t = 2 + 1.2 + 4.8

t = 8 hours