The volume of a cylinder varies jointly as the height of the cylinder and the square of its radius. If the height is halved and the radius is doubled, determine the effect on the volume.
halving the height, halves the volume
doubling the radius, quadruples the volume
1/2 * 4 = 2 ... volume is doubled
Let's denote the original height of the cylinder as h and the original radius as r. The original volume of the cylinder can be represented as V = k * h * r^2, where k is a constant of variation.
If the height is halved, the new height of the cylinder would be h/2.
If the radius is doubled, the new radius of the cylinder would be 2r.
The new volume of the cylinder can be represented as V' = k * (h/2) * (2r)^2.
Simplifying this expression, we have V' = k * (h/2) * 4r^2, which simplifies further to V' = k * h * 2r^2.
Comparing the new volume V' to the original volume V,
we can see that the new volume is twice the original volume.
Therefore, if the height is halved and the radius is doubled, the effect on the volume is that the volume doubles.
To determine the effect on the volume of a cylinder when the height is halved and the radius is doubled, we need to apply the concept of joint variation.
Let's denote the original height of the cylinder as h and the original radius as r. The volume of the original cylinder can be expressed as V = k * h * r², where k is the constant of variation.
Now, when the height is halved, we can represent the new height as h/2. Similarly, when the radius is doubled, we can represent the new radius as 2r.
The new volume of the cylinder can be represented as V' = k * (h/2) * (2r)²
Simplifying further, V' = k * (h/2) * 4r²
V' = 2k * h * 4r²
V' = 8k * h * r²
Comparing the new volume V' with the original volume V, we can see that the new volume V' is 8 times the original volume V.
Therefore, the effect on the volume of the cylinder when the height is halved and the radius is doubled is an increase by a factor of 8.