A spring has a natural length of 18 cm. If a 22-N force is required to keep it stretched to a length of 24 cm, how much work W is required to stretch it from 18 cm to 21 cm? (Round your answer to two decimal places.)

Take a look here. The first problem is just like this.

http://tutorial.math.lamar.edu/Classes/CalcI/Work.aspx

To find the work required to stretch the spring from 18 cm to 21 cm, we will use the formula for work:

W = 1/2 * k * (x2^2 - x1^2)

Where:
W is the work done
k is the spring constant
x2 is the final displacement
x1 is the initial displacement

First, let's find the spring constant, k. The spring constant represents the stiffness of the spring. To find k, we can use Hooke's Law, which states that the force required to stretch or compress a spring is directly proportional to the displacement.

F = k * x

Where:
F is the force applied
k is the spring constant
x is the displacement

In this case, we are given that a force of 22 N is required to stretch the spring to a length of 24 cm. We can convert this length to meters:

x = 24 cm = 0.24 m

Substituting the values into Hooke's Law, we have:

22 N = k * 0.24 m

Now, we can solve for k:

k = 22 N / 0.24 m
k ≈ 91.67 N/m

Now that we have the value of k, we can proceed to find the work done.

x2 = 21 cm = 0.21 m
x1 = 18 cm = 0.18 m

Substituting all the values into the formula for work, we have:

W = 1/2 * (91.67 N/m) * ((0.21 m)^2 - (0.18 m)^2)

W = 1/2 * 91.67 N/m * (0.0441 m^2 - 0.0324 m^2)

W = 1/2 * 91.67 N/m * 0.0117 m^2

W ≈ 0.538 J

Therefore, approximately 0.54 J of work is required to stretch the spring from 18 cm to 21 cm.