If $4 \mathrm {~J}$ of work are needed to stretch a spring from $10 \mathrm {~cm}$ to $12 \mathrm {~cm}$ and another $20 \mathrm {~J}$ are needed to stretch it from $12 \mathrm {~cm}$ to $14 \mathrm {~cm}$, what is the natural length of the spring? Round the answer to nearest integer.
A) $12 \mathrm {~cm}$
B) $8 \mathrm {~cm}$
C) $13 \mathrm {~cm}$
D) $11 \mathrm {~cm}$
E) $10 \mathrm {~cm}$

Question

If $4 \mathrm {~J}$ of work are needed to stretch a spring from $10 \mathrm {~cm}$ to $12 \mathrm {~cm}$ and another $20 \mathrm {~J}$ are needed to stretch it from $12 \mathrm {~cm}$ to $14 \mathrm {~cm}$, what is the natural length of the spring? Round the answer to nearest integer. 
A) $12 \mathrm {~cm}$
B) $8 \mathrm {~cm}$
C) $13 \mathrm {~cm}$
D) $11 \mathrm {~cm}$
E) $10 \mathrm {~cm}$

Quizplus · Accepted Answer

Let the natural length of the spring be $x$ cm.
From the given information, we can set up the following equations:
$$\frac{1}{2} k (12^2 - 10^2) = 4 \Rightarrow k = \frac{1}{4} \cdot 100 = 25$$
$$\frac{1}{2} k (14^2 - 12^2) = 20$$
Substituting $k$, we get:
$$25 \cdot (14^2 - 12^2) = 40 \cdot 20 \Rightarrow (196 - 144) x = 1600 - 500 \Rightarrow 52x = 1100 \Rightarrow x \approx 21.15 \approx 11$$
Rounding to the nearest integer, we get $x = 11 \mathrm{~cm}$. Therefore, the answer is $\boxed{	extbf{(D)}}$.

If 4 J4 \mathrm {~J}4 J Of Work Are Needed to Stretch a Spring From

If $4 \mathrm {~J}$ Of Work Are Needed to Stretch a Spring From