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 spring when no weight are attached.
I apologize for the mistake. Let's correct it.
Let the original length of the spring when no weights are attached be denoted by L inches.
We can set up two equations based on the given information:
When a 2-pound weight is attached:
L + (2/2) = 12
L + 1 = 12
L = 11
When a 5-pound weight is attached:
L + (5/2) = 18
L + 2.5 = 18
L = 15.5
Now we can find the rate of change in length with respect to weight:
(rate of change) = (15.5 - 11) / (5 - 2)
(rate of change) = 4.5 / 3
(rate of change) = 1.5
So, when no weight is attached:
L = 11 - 1.5(2)
L = 11 - 3
L = 8
Therefore, the length of the spring when no weights are attached is 8 inches.
Let's denote the original length of the spring when no weights are attached as L inches.
We can set up a proportion using the rate of change:
(rate of change) = (change in length) / (change in weight)
For the 2-pound weight:
(5-2) = (18-L)/(12-L)
For the 5-pound weight:
(5-2) = (18-L)/(12-L)
Now, we can solve the system of equations:
From the first equation: 3 = (18-L)/(12-L)
3(12-L) = 18-L
36-3L = 18-L
36 = 18+2L
18 = 2L
9 = L
Therefore, the length of the spring when no weights are attached is 9 inches.