A spring is manufactured with a pre-load. A force of 34.5 N compresses the spring by 1.5 cm. A force of 57.0 N stretches the same spring 0.03 m. Find the spring constant k (in N/m) and pre-load Fi (in N) that satisfies the linear equation describing this force, F = kx + Fi.

F = x +

To find the spring constant k and the pre-load Fi, we can use the given information about the spring:

1) A force of 34.5 N compresses the spring by 1.5 cm.
2) A force of 57.0 N stretches the spring by 0.03 m.

Let's use the information from (1) to find the spring constant k.

In the equation F = kx + Fi, when the spring is compressed, x will be negative. Therefore, the equation becomes:

34.5 = k(-0.015) + Fi

Simplifying the equation:

-0.015k + Fi = 34.5 (Equation 1)

Next, let's use the information from (2) to find the spring constant k (again) and the pre-load Fi.

When the spring is stretched, x will be positive. The equation becomes:

57.0 = k(0.03) + Fi

Simplifying the equation:

0.03k + Fi = 57.0 (Equation 2)

Now, we have a system of equations (Equation 1 and Equation 2). We can solve this system of equations simultaneously to find the values of k and Fi.

We can use any method to solve the system of equations. Let's use the substitution method to solve it here:

Step 1: Solve Equation 1 for Fi in terms of k:
Fi = 34.5 + 0.015k

Step 2: Substitute this expression for Fi into Equation 2:
0.03k + (34.5 + 0.015k) = 57.0

Step 3: Simplify and solve for k:
0.045k + 34.5 = 57.0
0.045k = 57.0 - 34.5
0.045k = 22.5
k = 22.5 / 0.045
k = 500 N/m

Step 4: Substitute the value of k back into Equation 1 to find Fi:
Fi = 34.5 + 0.015(500)
Fi = 34.5 + 7.5
Fi = 42 N

Therefore, the spring constant k is 500 N/m, and the pre-load Fi is 42 N.