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

### Precalculus: Mathematics for Calcu...

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

### 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

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 5−lb of weight is attached.

Expert Solution

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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