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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

A high-speed bullet train accelerates and decelerates at the rate of 4 ft/s2. Its maximum cruising speed is 90 mi/h.

(a) What is the maximum distance the train can travel if it accelerates from rest until it reaches its cruising speed and then runs at that speed for 15 minutes?

(b) Suppose that the train starts from rest and must come to a complete stop in 15 minutes. What is the maximum distance it can travel under these conditions?

(c) Find the minimum time that the train takes to travel between two consecutive stations that are 45 miles apart.

(d) The trip from one station to the next takes 37.5 minutes. How far apart are the stations?

(a)

To determine

The maximum distance travelled by bullet train, if it accelerates from rest until it reaches its cruising speed.

Explanation

Given data:

The rate of deceleration (a(t)) is 4fts2 and maximum cruising speed (v(t)) is 90mih.

Formula Used:

Write the expression for acceleration function a(t).

a(t)=v(t) (1)

Here,

v(t) is first derivative of velocity function v(t).

Write the expression for velocity function v(t).

v(t)=s(t) (2)

Here,

s(t) is first derivative of displacement function s(t).

Antiderivative of tn is tn+1n+1 and 1 is t.

Calculation:

Initially the bullet train is at rest, that is v(0)=0 and consider the initial displacement as zero, that is s(0)=0.

Substitute 4fts2 for a(t) in equation (1),

v(t)=4

Antiderivate the expression with respect to t.

v(t)=4t+C (3)

Here,

C is arbitrary constant.

Substitute 0 for t in equation (3),

v(0)=4(0)+C=C

Substitute 0 for v(0),

C=0

Substitute 0 for C in equation (3),

v(t)=4t+0

v(t)=4t (4)

Substitute 4t for v(t) in equation (2),

s(t)=4t

Antiderivate the expression with respect to t.

s(t)=4(12t2)+D

s(t)=2t2+D (5)

Here,

D is arbitrary constant.

Substitute 0 for t in equation (5),

s(0)=2(02)+D=0+D=D

Substitute 0 for s(0),

D=0

Substitute 0 for D in equation (5),

s(t)=2t2+0

s(t)=2t2 (6)

Substitute 90mih for v(t) in equation (2),

90mih=4t4t=90(52803600)fts

(b)

To determine

The maximum distance travelled by train.

(c)

To determine

To find: The minimum time that train takes to travel between two consecutive stations which are 45 miles apart.

(d)

To determine

To find: The distance between the stations.

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