To determine the solution, let's follow through step-by-step.
### Step 1: Determine the proportionality constant [tex]\( k \)[/tex]
We know from Hooke's Law that the force [tex]\( F \)[/tex] needed to stretch a spring by distance [tex]\( x \)[/tex] is proportional:
[tex]\[ F = kx \][/tex]
Where [tex]\( k \)[/tex] is the proportionality constant.
Given:
- [tex]\( F_1 = 180 \)[/tex] newtons
- [tex]\( x_1 = 6 \)[/tex] cm
We can find [tex]\( k \)[/tex] by rearranging the equation:
[tex]\[ k = \frac{F_1}{x_1} \][/tex]
Substitute the known values:
[tex]\[ k = \frac{180 \text{ N}}{6 \text{ cm}} = 30 \text{ N/cm} \][/tex]
So, the direct variation equation for the spring is:
[tex]\[ F = 30x \][/tex]
Thus, answer b: [tex]\( F = 30x \)[/tex] is correct.
### Step 2: Determine the distance stretched for a new force
Given a new force:
- [tex]\( F_2 = 420 \)[/tex] newtons
Using the direct variation equation [tex]\( F = 30x \)[/tex]:
[tex]\[ 420 = 30x \][/tex]
To find [tex]\( x_2 \)[/tex], divide both sides by 30:
[tex]\[ x = \frac{420}{30} = 14 \text{ cm} \][/tex]
Thus, the distance the spring will be stretched under a force of 420 newtons is 14 cm.
So, answer d: [tex]\( x = 14 \)[/tex] is correct.
### Conclusion:
The correct answers are:
- [tex]\( F = 30 x \)[/tex]
- [tex]\( x = 14 \)[/tex]
