To solve the problem, we can use the direct variation relationship between the force (F) and the change in length (x) of the spring. This relationship can be expressed as:
\[ F = kx \]
where \( k \) is the constant of proportionality.
Given that a force of 20 lb compresses the spring by 4 inches, we can find \( k \):
\[ 20 = k \cdot 4 \]
To find \( k \), divide both sides by 4:
\[ k = \frac{20}{4} = 5 , \text{lb/in.} \]
Now that we have \( k \), we can use it to find the force required to compress the spring by 7 inches.
Using the same formula:
\[ F = kx \]
Substituting \( k = 5 \) lb/in. and \( x = 7 \) in:
\[ F = 5 \cdot 7 = 35 , \text{lb.} \]
Therefore, the force required to compress the spring 7 inches is 35 lb.