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

Answer:  25 miles

Step-by-step explanation:

Since, the total distance = 31 miles.

Let he walks x miles at the rate of 4 miles per hour.

⇒ he walks remaining (31-x) miles at the rate of 50 miles per hour.

According to the question,

$\frac{x}{4} + \frac{31-x}{50} = 2$

⇒ $\frac{25x}{100} + \frac{62-2x}{100} = 2$

⇒  $\frac{25x+62-2x}{100}= 2$

⇒  $\frac{23x+62}{100}= 2$

⇒ $23 x + 62 = 200$

⇒ $23 x = 200 - 62$ ( by subtracting 62 on both sides )

⇒  $23 x = 138$

⇒ $x=\frac{138}{23}$ ( Dividing both sides by 23 )

⇒ $x = 6$

Thus, the distance he pays to the cab driver = 31-x= 31 - 6 = 25 miles.

Therefore, He pays to the cab driver for 25 miles.

Answer 2

Distance that must be paid to the 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

so

$\rm t=\dfrac{x}{v}$

A mountain path is 31 miles long

The distance traveled by tourists = x1

The distance traveled taxi = x2, such that

x1 + x2 = 31

x1 = 31-x2

total travel time (t) = 2 hours

tourist speed = v1 = 4 mph

taxi speed = v2 = 50 mph

time taken by tourists = t1

$\rm \dfrac{x1}{v1}=\dfrac{x1}{4}$

the time taken by taxi = t2

$\rm \dfrac{x2}{v2}=\dfrac{x2}{50}$

total time = t1 + t2

$\rm 2=\dfrac{x1}{4}+\dfrac{x2}{50}\\\\2=\dfrac{25x1+2x2}{100}\\\\200=25(31-x2)+2x2\\\\200=775-25x2+2x2\\\\575=23x2\\\\x2=\boxed{\bold{25~miles}}$

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