Hooke's Law the table shows the distance D that a spring stretches when a weight W is hung on it.
W (pounds 2 6 9 15
D (inches) 1.6 4.8 7.2 12
(a.) Do the data represent direct or inverse variation?
Explain. (b.) Find an equation that models the data.
(c.) How far will the spring stretc if an 11- pound weight is hung on it, as depicted in the figure?
(a) when the ratios are proportional they represent direct variation.
For example: 1.5/2 = 4.8/6 = 7.2/9 = 12/15 = 0.8
Guess at this point you can figure out the equation required for (b).
(c) substitute W=11 into the equation in (b).
Post for checking if you wish.
To determine if the given data represents direct or inverse variation, we need to examine the relationship between the two variables: weight W (in pounds) and distance D (in inches).
(a.) Direct Variation: If the data represents direct variation, then the ratio of W to D should remain constant. To test this, we can calculate the ratios for each pair of weights W and distances D:
- For W = 2 pounds, D = 1.6 inches, so W/D ≈ 1.25
- For W = 6 pounds, D = 4.8 inches, so W/D ≈ 1.25
- For W = 9 pounds, D = 7.2 inches, so W/D ≈ 1.25
- For W = 15 pounds, D = 12 inches, so W/D ≈ 1.25
Since the ratios of W/D for all weights are approximately equal to 1.25, the data represents direct variation.
Explanation: In direct variation, if one variable (W) increases by a constant factor, then the other variable (D) also increases by the same constant factor. In this case, as the weight increases, the distance stretched by the spring also increases, but in the same ratio.
(b.) To find an equation that models the data, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the distance stretched. Mathematically, Hooke's Law can be represented as:
F = k * x
Where F is the force (weight) applied to the spring, k is the spring constant, and x is the distance stretched by the spring.
Here, we can equate weight W to force F, and distance D to x. So the equation becomes:
W = k * D
To find the value of k, we can use any pair of values from the given data. Let's choose W = 2 pounds and D = 1.6 inches:
2 = k * 1.6
Solving for k, we get k ≈ 1.25
Therefore, the equation that models the data is:
W = 1.25D
(c.) To find out how far the spring will stretch when an 11-pound weight is hung on it, we can substitute W = 11 into the equation:
11 = 1.25D
Solving for D, we get:
D ≈ 8.8 inches
Therefore, when an 11-pound weight is hung on the spring, it will stretch approximately 8.8 inches.