BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071
BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

Solutions

Chapter 1, Problem 131RE

a.

To determine

To calculate:For the given equation which depicts Hooke’s law for a given spring, given by s=0.3w+2.5 the objective is to find out what does the slope and s -intercept represent in this equation.

Expert Solution

Answer to Problem 131RE

In the equation s=0.3w+2.5 the slope represents the stretch in length per unit weight increase and the s -intercept represents stretch in length when no weight is attached to the spring.

Explanation of Solution

Given information:

Hooke’s law equation for a given spring, given by s=0.3w+2.5

Formula used:

Hooke’s Law precisely states that force required for a given spring to keep it stretched x units beyond its original lent is actually directly proportional to x itself and the constant of proportionality uses here is referred as the spring constant.

It can be represented as the following equation:

  Fs=kx

Where Fs represents the force on spring, k is the spring constant and x is the stretch or the compression made in the spring

Slope-intercept equation for a given line which has slope as m and y −intercept as b is:

  y=mx+b

Slope m of the line passing through two points in general say P1=(x1,y1) and P2=(x2,y2) is:

  m=y2y1x2x1

Point-slope equation for a given line passing along point P1=(x1,y1) is:

  yy1=m(xx1)

Calculation:

Consider theHooke’s law equation for a given spring, given by s=0.3w+2.5

Recall that Hooke’s Law precisely states that force required for a given spring to keep it stretched x units beyond its original lent is actually directly proportional to x itself and the constant of proportionality uses here is referred as the spring constant.

It can be represented as the following equation:

  Fs=kx

Where Fs represents the force on spring, k is the spring constant and x is the stretch or the compression made in the spring

Therefore, the equation s=0.3w+2.5 means that the weight w is attached to a spring which is hanging and then by Hooke’s law the stretched length for the given spring is directly proportional to w .

Now recall, Slope-intercept equation for a given line which has slope as m and y −intercept as b is:

  y=mx+b

Therefore, comparing this with s=0.3w+2.5 to get:

Slope m=0.3

And

  y -intercept or the s -intercept as 2.5

The slope m=0.3 basically represents the increase in the stretched length of the given spring per unit the increase in the weight hanging along the spring.

And, the s -intercept 2.5 for the required equation of the spring denotes the stretched length in the case when there is no weight attached to the given spring.

b.

To determine

To calculate:For the given equation which depicts Hooke’s law for a given spring, given by s=0.3w+2.5 the objective is to find out what will be the length of the spring when a 5lb of weight is attached.

Expert Solution

Answer to Problem 131RE

The length of the spring when 5lb of weight is attached is 4 inches.

Explanation of Solution

Given information:

Hooke’s law equation for a given spring, given by s=0.3w+2.5

Formula used:

Hooke’s Law precisely states that force required for a given spring to keep it stretched x units beyond its original length is actually directly proportional to x itself and the constant of proportionality uses here is referred as the spring constant.

It can be represented as the following equation:

  Fs=kx

Where Fs represents the force on spring, k is the spring constant and x is the stretch or the compression made in the spring

Slope-intercept equation for a given line which has slope as m and y −intercept as b is:

  y=mx+b

Slope m of the line passing through two points in general say P1=(x1,y1) and P2=(x2,y2) is:

  m=y2y1x2x1

Point-slope equation for a given line passing along point P1=(x1,y1) is:

  yy1=m(xx1)

Calculation:

Consider the Hooke’s law equation for a given spring, given by s=0.3w+2.5

Recall that Hooke’s Law precisely states that force required for a given spring to keep it stretched x units beyond its original length is actually directly proportional to x itself and the constant of proportionality uses here is referred as the spring constant.

It can be represented as the following equation:

  Fs=kx

Where Fs represents the force on spring, k is the spring constant and x is the stretch or the compression made in the spring

Therefore, the equation s=0.3w+2.5 means that the weight w is attached to a spring which is hanging and then by Hooke’s law the stretched length for the given spring is directly proportional to w .

Now recall, Slope-intercept equation for a given line which has slope as m and y −intercept as b is:

  y=mx+b

Therefore, comparing this with s=0.3w+2.5 to get:

Slope m=0.3

And

  y -intercept or the s -intercept as 2.5

Let s be thelength of the spring when 5lb of weight is attached.

Therefore,

  s=0.3w+2.5

Put w=5 to get:

  s=0.3(5)+2.5s=1.5+2.5s=4

Hence, the length of the spring when 5lb of weight is attached is 4 inches.

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