Use the image to answer the question.
An illustration shows three coiled wire springs stretched to varying lengths. The first spring does not have any weights pulling it downward, and the length of the spring is marked as a question mark inches. The second spring has a 2-pound weight attached to the bottom, and its length is marked as 12 inches. The third spring has a 5-pound weight attached to the bottom and its length is marked as 18 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.
(1 point)
inches
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 represents how the length of the spring changes with respect to the weight applied.
Let's consider the second spring, which has a length of 12 inches when a 2-pound weight is attached. We can represent this relationship using the formula:
Length = k * Weight + c
Where "Length" represents the length of the spring, "Weight" represents the weight attached to it, "k" represents the rate of change, and "c" represents a constant term.
Using the given values, we can substitute them into the formula:
12 = 2k + c ...(Equation 1)
Now let's consider the third spring, which has a length of 18 inches when a 5-pound weight is attached. We can use the same formula:
18 = 5k + c ...(Equation 2)
We now have a system of two linear equations with two variables (k and c). We can solve this system of equations to find the values of k and c.
Subtracting Equation 1 from Equation 2, we get:
18 - 12 = (5k + c) - (2k + c)
6 = 3k
Dividing both sides by 3, we find:
k = 2
Now, substitute the value of k into either Equation 1 or Equation 2:
12 = 2(2) + c
12 = 4 + c
Subtracting 4 from both sides, we find:
c = 8
Therefore, the equation describing the relationship between the length of the spring and the weight attached is:
Length = 2 * Weight + 8
To find the length of the spring when no weights are attached, we can substitute a weight of 0 into the equation:
Length = 2 * 0 + 8
Length = 8 inches
Hence, the length of the spring when no weights are attached is 8 inches.
ARE YOU SURE
To find the length of the spring when no weights are attached, we need to determine the rate of change of the length of the spring with respect to the weight.
First, we can calculate the rate of change using the information given. The length of the spring increases by 6 inches when the weight increases from 2 pounds to 5 pounds.
Rate of change = (change in length)/(change in weight) = (18 - 12)/(5 - 2) = 6/3 = 2 inches per pound.
Since the length of the spring increases by 2 inches for every pound of weight, we can determine the length of the spring when no weights are attached by subtracting 2 inches from the length of the spring when a 2-pound weight is attached.
Length of the spring without weights = 12 inches - 2 inches = 10 inches.
Therefore, the length of the spring when no weights are attached is 10 inches.