Answer :
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]
[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]