Let's represent the length of the spring when no weights are attached as $x$.
We can find the rate of change of the spring's length by calculating the change in length per pound of weight.
The rate of change when going from a 2-pound weight to a 5-pound weight is:
\begin{align*}
\text{Rate of change} &= \frac{\text{Change in length}}{\text{Change in weight}}\\
&= \frac{18-12}{5 - 2}\\
&= \frac{6}{3}\\
&= 2 \text{ inches/pound}
\end{align*}
To find the length when no weights are attached, we can use this rate of change. Since the change in weight is from 2 pounds to no weight (0 pounds), the change in length can be calculated as:
\begin{align*}
\text{Change in length} &= \text{Rate of change} \times \text{Change in weight}\\
&= 2 \times (2 - 0)\\
&= 4 \text{ inches}
\end{align*}
Therefore, the length of the spring when no weights are attached is $12 + 4 = \boxed{16}$ inches.
A spring has a length of 12 inches when a 2-pound weight is attached, and a length of 18 inches when a 5-pound weight is attached. Use rate of change to find the length of the spring when no weights are attached
3 answers
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Apologies for the mistake. Let's recalculate.
Given that the spring has a length of 12 inches when a 2-pound weight is attached and a length of 18 inches when a 5-pound weight is attached, we can find the rate of change of the spring's length.
The rate of change when going from a 2-pound weight to a 5-pound weight is:
\begin{align*}
\text{Rate of change} &= \frac{\text{Change in length}}{\text{Change in weight}}\\
&= \frac{18-12}{5 - 2}\\
&= \frac{6}{3}\\
&= 2 \text{ inches/pound}
\end{align*}
To find the length when no weights are attached, we can use this rate of change. Since the change in weight is from 2 pounds to no weight (0 pounds), the change in length can be calculated as:
\begin{align*}
\text{Change in length} &= \text{Rate of change} \times \text{Change in weight}\\
&= 2 \times (0 - 2)\\
&= -4 \text{ inches}
\end{align*}
Therefore, the length of the spring when no weights are attached is $12 + (-4) = \boxed{8}$ inches.
Given that the spring has a length of 12 inches when a 2-pound weight is attached and a length of 18 inches when a 5-pound weight is attached, we can find the rate of change of the spring's length.
The rate of change when going from a 2-pound weight to a 5-pound weight is:
\begin{align*}
\text{Rate of change} &= \frac{\text{Change in length}}{\text{Change in weight}}\\
&= \frac{18-12}{5 - 2}\\
&= \frac{6}{3}\\
&= 2 \text{ inches/pound}
\end{align*}
To find the length when no weights are attached, we can use this rate of change. Since the change in weight is from 2 pounds to no weight (0 pounds), the change in length can be calculated as:
\begin{align*}
\text{Change in length} &= \text{Rate of change} \times \text{Change in weight}\\
&= 2 \times (0 - 2)\\
&= -4 \text{ inches}
\end{align*}
Therefore, the length of the spring when no weights are attached is $12 + (-4) = \boxed{8}$ inches.