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
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.
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.
are you sure?
To find the length of the spring when no weights are attached, we can use the concept of rate of change. The rate of change is a measure of how one quantity is changing with respect to another quantity.
In this case, we have the length of the spring changing with respect to the weight attached to it. Let's define the length of the spring as L (in inches) and the weight as W (in pounds). We are given two data points:
When W = 2 pounds, L = 12 inches
When W = 5 pounds, L = 18 inches
To find the rate of change, we need to calculate the change in length of the spring divided by the change in weight.
Change in Length = L2 - L1
Change in Weight = W2 - W1
Substituting the given values, we have:
Change in Length = 18 - 12 = 6 inches
Change in Weight = 5 - 2 = 3 pounds
Now, we can calculate the rate of change by dividing the change in length by the change in weight:
Rate of Change = Change in Length / Change in Weight = 6 inches / 3 pounds
This gives us a rate of change of 2 inches per pound.
To find the length of the spring when no weights are attached, we can use this rate of change. Since the length of the spring increases by 2 inches for each pound, when no weights are attached (i.e., W = 0 pounds), the length of the spring would be:
Length = 12 inches + (2 inches/pound) * 0 pounds
Length = 12 inches
Therefore, the length of the spring when no weights are attached is 12 inches.