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.
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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 the amount of change in one variable (in this case, the length of the spring) for a given change in another variable (in this case, the weight attached to the spring).
From the given information, we can see that when the weight increases from 2 pounds to 5 pounds, the length of the spring increases from 12 inches to 18 inches. This means that for every 3-inch increase in the length of the spring, there is a 3-pound increase in the weight.
To find the length of the spring when no weights are attached, we can determine how much the length would increase for a 0-pound increase in weight. Since the rate of change is consistent, we can set up a proportion:
(3 inches)/(3 pounds) = (0 inches)/(0 pounds)
We can solve this proportion by cross-multiplying:
3 inches * 0 pounds = 3 pounds * 0 inches
0 = 0
This tells us that when there is no increase in weight, there is also no increase in the length of the spring. Therefore, the length of the spring when no weights are attached is the same as the length of the spring marked as a question mark, which is 12 inches.
Answer: The length of the spring when no weights are attached is 12 inches.
is this true?
Apologies, I made a mistake in my previous response. Thank you for catching that.
To find the length of the spring when no weights are attached, we can use the concept of rate of change.
From the given information, we can see that when the weight increases from 2 pounds to 5 pounds, the length of the spring increases from 12 inches to 18 inches. This means that for every 3-pound increase in weight, there is a 6-inch increase in the length of the spring.
To find the length of the spring when no weights are attached, we can determine how much the length would increase for a 0-pound increase in weight. Since the rate of change is consistent, we can set up a proportion:
(3 pounds)/(6 inches) = (0 pounds)/(x inches)
We can solve this proportion by cross-multiplying:
3 pounds * x inches = 6 inches * 0 pounds
3x = 0
x = 0
Therefore, the length of the spring when no weights are attached is 0 inches.
Answer: The length of the spring when no weights are attached is 0 inches.
still wrong
Apologies once again for the incorrect response. Let's reassess the problem using the correct information.
From the given information, we are told that the length of the spring is 12 inches when a 2-pound weight is attached and 18 inches when a 5-pound weight is attached.
To find the length of the spring when no weights are attached, we need to determine the rate of change in length for a given change in weight. In this case, the rate of change is the same regardless of the specific weight attached.
We can set up a proportion to solve for the length of the spring when no weights are attached:
(Change in length)/(Change in weight) = (12 inches - ? inches)/(2 pounds - 0 pounds) = (18 inches - ? inches)/(5 pounds - 0 pounds)
Simplifying the proportion:
(12 - ?)/2 = (18 - ?)/5
Cross-multiplying:
5(12 - ?) = 2(18 - ?)
60 - 5? = 36 - 2?
-5? + 2? = 36 - 60
-3? = -24
? = (-24)/(-3)
? = 8
Therefore, the length of the spring when no weights are attached is 8 inches.
Answer: The length of the spring when no weights are attached is 8 inches.