The volume of a cylinder varies jointly as the height of the cylinder and the square of its radius. If the height is doubled and the radius is halved, determine the effect on the volume.
v = π * r^2 * h
v = π * (r/2)^2 * 2 h = π * [(r^2) / 4] * 2 h
To determine the effect on the volume of a cylinder when the height is doubled and the radius is halved, we can use the concept of joint variation.
Let's denote the original volume as V and the original height and radius as h and r, respectively.
The volume of a cylinder is given by the formula: V = πr^2h
According to the problem, volume (V) varies jointly with the height (h) and the square of the radius (r^2). This can be expressed as:
V = kh * r^2
where k is a constant of variation.
Now, let's determine the effect of doubling the height and halving the radius on the volume.
New height = 2h (height is doubled)
New radius = r/2 (radius is halved)
Substituting these new values into the formula, we get:
New volume (V') = k(2h) * (r/2)^2
= 2kh * (r^2/4)
= (1/2) * (2kh * r^2)
= (1/2) * V
Therefore, the effect of doubling the height and halving the radius on the volume is that the new volume becomes half of the original volume.
In other words, the new volume is reduced to half of the original volume when the height is doubled and the radius is halved.
To determine the effect on the volume when the height is doubled and the radius is halved, we need to understand how the volume of a cylinder is related to its height and radius.
We can start by writing the equation that relates the volume (V), height (h), and radius (r) of a cylinder. The problem states that the volume varies jointly with the height and the square of the radius, which can be written as:
V = k * h * r^2
Here, k is a constant of variation. The value of k will depend on the specific units used for measuring the volume, height, and radius.
Now, let's consider the effect of doubling the height and halving the radius on the volume. We will use the variables h1, r1, and V1 to represent the initial values of height, radius, and volume, and h2, r2, and V2 to represent the new values after the changes.
Given the changes:
h2 = 2 * h1 (doubling the height)
r2 = (1/2) * r1 (halving the radius)
Substituting these values into the equation for the volume, we get:
V2 = k * h2 * r2^2
= k * (2 * h1) * ((1/2) * r1)^2
= k * 2 * h1 * (1/4) * r1^2
= (1/2) * k * h1 * r1^2
Notice that V2 is equal to half of V1:
V2 = (1/2) * V1
Therefore, when the height is doubled and the radius is halved, the effect on the volume is that it becomes half of the original volume.