Question

A tourist starts to walk up a mountain path that is 31 miles long at the rate of 4 miles per hour. After walking for a while, he gets tired and decides to get a taxi. The taxi gets him to the top travelling at a constant speed of 50 mph. If the tourist reaches the destination 2 hours after he started, what distance does he have to pay the cab driver for?

Answer 1

Let the Distance traveled by the tourist by walking be : D miles

Given : Tourist walks at a rate of 4 Miles per Hour

$\mathsf{\implies Tourist\;travels\;D\;Miles\;in : \frac{D}{4}\;Hours}$

Given : The Total Distance of Mountain Path = 31 Miles

Distance traveled by the tourist by hiring a Taxi is : (31 - D) Miles

Given : The Taxi travels at a constant speed of 50 miles per hour

$\mathsf{\implies Tourist\;travels\;(31 - D)\;Miles\;in : (\frac{31 - D}{50})\;Hours}$

Given : The Tourist reaches the destination 2 Hours after he started

Time taken by Walking + Time Taken by travelling in Taxi = 2 Hours

$\mathsf{\implies \frac{D}{4} + \frac{31 - D}{50} = 2}$

$\mathsf{\implies \frac{D}{2} + \frac{31 - D}{25} = 4}$

$\mathsf{\implies (\frac{25D + 2(31 - D)}{50}) = 4}$

$\mathsf{\implies (\frac{25D + 62 - 2D}{50}) = 4}$

$\mathsf{\implies ({23D + 62}) = 200}$

$\mathsf{\implies 23D = 200 - 62}$

$\mathsf{\implies 23D = 138}$

$\mathsf{\implies D = 6}$

⇒ Distance traveled by the tourist by walking = 6 Miles

⇒ Distance traveled by the tourist by hiring a taxi = (31 - 6) = 25 Miles

⇒ The Distance for which tourist has to pay for cab driver is 25 Miles

Answer 2

The distance traveled by taxi = 25 miles

Further explanation

Linear motion consists of 2: constant velocity motion with constant velocity and uniformly accelerated motion with constant acceleration

• At constant velocity motion:

the speed of vo = v = constant

acceleration = a = 0

Δx = vt or x = xo + vt

An equation of constant velocity motion

$\large {\boxed {\bold {x = v \times \: t}}}$

x = distance = m

v = speed = m / s

t = time = seconds

x1, x2 = distance traveled

t1, t2 = travel time

1 = tourist, 2 = taxi

$\rm x1=v1.t1=4t1\\\\x2=v2.t2=50.t2\\\\t1+t2=2~hours\\\\t1=2-t2$

Total distance = x1 + x2

$\rm 31~miles=4t1+50t2\\\\31=4(2-t2)+50t2\\\\31=8-4t2+50t2\\\\23=46t2\\\\t2=0.5$

then:

$\rm x2=v2.t2\\\\x2=50\times 0.5\\\\x2=\boxed{\bold{25~miles}}$

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