To find the force [tex]\( F \)[/tex] that will produce a length of 14 units for the spring, we start with the given equation:
[tex]\[
L = \frac{2}{3} F + 12
\][/tex]
We are given that the length [tex]\( L \)[/tex] is 14, so we can substitute 14 for [tex]\( L \)[/tex] in the equation:
[tex]\[
14 = \frac{2}{3} F + 12
\][/tex]
Next, we need to isolate [tex]\( F \)[/tex] on one side of the equation. First, subtract 12 from both sides:
[tex]\[
14 - 12 = \frac{2}{3} F
\][/tex]
This simplifies to:
[tex]\[
2 = \frac{2}{3} F
\][/tex]
Now, to solve for [tex]\( F \)[/tex], we multiply both sides by the reciprocal of [tex]\( \frac{2}{3} \)[/tex], which is [tex]\( \frac{3}{2} \)[/tex]:
[tex]\[
F = 2 \times \frac{3}{2}
\][/tex]
This simplifies to:
[tex]\[
F = 3
\][/tex]
Therefore, the force [tex]\( F \)[/tex] that will produce a length of 14 units is:
[tex]\[
F = 3.0
\][/tex]
