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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 attach...Asked by Demon
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)
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Answered by
GPT 3.5
Let x represent the length of the spring when no weights are attached.
Let y represent the weight in pounds.
We are 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. This can be represented by the following two equations:
12 = x + 2 (equation 1)
18 = x + 5 (equation 2)
First, let's solve equation 1 for x by subtracting 2 from both sides:
12 - 2 = x + 2 - 2
10 = x
Next, let's solve equation 2 for x by subtracting 5 from both sides:
18 - 5 = x + 5 - 5
13 = x
Since we have two different values for x, there is an error in our calculations. Let's check our work:
When a 2-pound weight is attached, the length of the spring is 12 inches. This means that 12 = x + 2. Using x = 10:
12 = 10 + 2
12 = 12
This checks out.
When a 5-pound weight is attached, the length of the spring is 18 inches. This means that 18 = x + 5. Using x = 13:
18 = 13 + 5
18 = 18
This checks out as well.
Therefore, the length of the spring when no weights are attached is x = 10 inches.
Let y represent the weight in pounds.
We are 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. This can be represented by the following two equations:
12 = x + 2 (equation 1)
18 = x + 5 (equation 2)
First, let's solve equation 1 for x by subtracting 2 from both sides:
12 - 2 = x + 2 - 2
10 = x
Next, let's solve equation 2 for x by subtracting 5 from both sides:
18 - 5 = x + 5 - 5
13 = x
Since we have two different values for x, there is an error in our calculations. Let's check our work:
When a 2-pound weight is attached, the length of the spring is 12 inches. This means that 12 = x + 2. Using x = 10:
12 = 10 + 2
12 = 12
This checks out.
When a 5-pound weight is attached, the length of the spring is 18 inches. This means that 18 = x + 5. Using x = 13:
18 = 13 + 5
18 = 18
This checks out as well.
Therefore, the length of the spring when no weights are attached is x = 10 inches.
Answered by
GPT 3.5
Apologies for the confusion. Let's reassess the problem.
Let x represent the length of the spring when no weights are attached.
Let y represent the weight in pounds.
We are 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. This can be represented by the following two equations:
12 = x + 2 (equation 1)
18 = x + 5 (equation 2)
To find the rate of change, we need to calculate the difference in the length of the spring for a given difference in weight. From equations 1 and 2, we can establish the rate of change as:
Rate of change = (change in length)/(change in weight)
Substituting the given values into the rate of change equation:
Rate of change = (18 - 12)/(5 - 2)
= 6/3
= 2 inches
This means that for every 1-pound increase in weight, the length of the spring increases by 2 inches.
To find the length of the spring when no weights are attached (0 pounds), we can apply the rate of change:
Length of the spring when no weights are attached = 12 - 2(0)
= 12 inches.
Therefore, the length of the spring when no weights are attached is 12 inches.
Let x represent the length of the spring when no weights are attached.
Let y represent the weight in pounds.
We are 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. This can be represented by the following two equations:
12 = x + 2 (equation 1)
18 = x + 5 (equation 2)
To find the rate of change, we need to calculate the difference in the length of the spring for a given difference in weight. From equations 1 and 2, we can establish the rate of change as:
Rate of change = (change in length)/(change in weight)
Substituting the given values into the rate of change equation:
Rate of change = (18 - 12)/(5 - 2)
= 6/3
= 2 inches
This means that for every 1-pound increase in weight, the length of the spring increases by 2 inches.
To find the length of the spring when no weights are attached (0 pounds), we can apply the rate of change:
Length of the spring when no weights are attached = 12 - 2(0)
= 12 inches.
Therefore, the length of the spring when no weights are attached is 12 inches.