The coiled spring of a toy directly supports the weight of a child. The spring is compressed a distance of 2.9 inches by the weight of a 30-pound child. The toy will not work properly if its spring is compressed more than 4 inches. What is the value of the constant of variation? What is the weight of the heaviest child who should be allowed to use the toy?
F = kx, so
30 = 2.9k
max weight is 4k
To find the constant of variation, we need to use the formula for Hooke's Law, which is F = kx, where F is the force applied, k is the constant of variation, and x is the displacement of the spring.
In this case, the weight of the child applies the force, and the displacement of the spring is the compression distance.
Given that the spring is compressed a distance of 2.9 inches by a 30-pound child, we can write this as:
30 = k * 2.9
To find the value of the constant of variation (k), we divide both sides of the equation by 2.9:
k = 30 / 2.9 ≈ 10.34
Therefore, the value of the constant of variation is approximately 10.34.
Now let's find the weight of the heaviest child who should be allowed to use the toy without compressing the spring more than 4 inches.
To do this, we rearrange the formula:
F = kx
Solving for x:
x = F / k
Substituting the values, with x equal to 4 inches and k equal to 10.34:
4 = F / 10.34
Rearranging the equation to solve for F:
F = 4 * 10.34
F ≈ 41.36
Therefore, the weight of the heaviest child who should be allowed to use the toy is approximately 41.36 pounds.
To find the value of the constant of variation, we can use the formula for the variation of direct proportionality.
The formula for direct variation can be written as: y = kx, where y is the dependent variable, x is the independent variable, and k is the constant of variation.
In this case, the dependent variable is the compression distance of the spring (y) and the independent variable is the weight of the child (x).
Using the given information, we know that when the weight of a 30-pound child compresses the spring, the distance of compression is 2.9 inches. Therefore, we can write this information in the form of the equation:
2.9 = k * 30
To find the value of k, we divide both sides of the equation by 30:
k = 2.9 / 30
Calculating this gives us:
k ≈ 0.0967
So, the value of the constant of variation (k) is approximately 0.0967.
To find the weight of the heaviest child who should be allowed to use the toy, we can use the maximum compression limit given.
When the spring is compressed by the maximum limit of 4 inches, we need to calculate the weight (x) that corresponds to this compression distance (y). Using the equation:
4 = 0.0967 * x
To solve for x, we divide both sides of the equation by 0.0967:
x = 4 / 0.0967
Calculating this gives us:
x ≈ 41.35
Therefore, the weight of the heaviest child who should be allowed to use the toy is approximately 41.35 pounds.