How does the volume of a cone change when the radius is quadrupled and the height is reduced to 2004-06-01-04-00_files/i0300000.jpg of its original size
what is the answer
Well, if the radius is quadrupled and the height is reduced to 1/4 of its original size, it's safe to say that the volume of the cone will be quatro-much smaller! Or in other words, it will be one-sixteenth of its original volume. So, prepare for some cone-venient shrinkage!
To determine how the volume of a cone changes when the radius is quadrupled and the height is reduced to 1/4 of its original size, we can use the formula for the volume of a cone.
The formula for the volume of a cone is: V = (1/3)πr^2h, where V is the volume, π is a mathematical constant approximately equal to 3.14159, r is the radius, and h is the height.
Let's assume the original radius of the cone is represented by 'r' and the original height is represented by 'h'.
When the radius is quadrupled, the new radius will be 4r.
When the height is reduced to 1/4 of its original size, the new height will be h/4.
Using the formula for the volume of a cone, we can calculate the new volume:
New Volume = (1/3)π(4r)^2(h/4)
= (1/3)π(16r^2)(h/4)
= (1/3)π(4r^2)(h)
= (4/3)πr^2h
Comparing the new volume (V_new) to the original volume (V_orig), we get:
V_new = (4/3)πr^2h
V_orig = (1/3)πr^2h
The ratio of the new volume to the original volume is:
V_new/V_orig = [(4/3)πr^2h] / [(1/3)πr^2h]
= 4
Therefore, when the radius is quadrupled and the height is reduced to 1/4 of its original size, the volume of the cone increases by a factor of 4.
To determine how the volume of a cone changes when the radius is quadrupled and the height is reduced to 1/3 of its original size, we can use the formula for the volume of a cone: V = (1/3)πr^2h, where V is the volume, π is approximately 3.14, r is the radius, and h is the height.
Let's denote the original radius as r1 and the original height as h1. The original volume can be expressed as V1 = (1/3)πr1^2h1.
When the radius is quadrupled, the new radius becomes 4r1. This means the radius has increased by a factor of 4. Similarly, when the height is reduced to 1/3 of its original size, the new height becomes (1/3)h1. This implies the height has decreased by a factor of 3.
Substituting the new values into the volume formula, we have V2 = (1/3)π(4r1)^2((1/3)h1). Simplifying this expression, we get V2 = (1/3)π(16r1^2)(1/3h1), which further simplifies to V2 = (16/9)πr1^2h1.
Comparing the new volume (V2) to the original volume (V1), we can express the ratio of the new volume to the original volume as V2/V1 = [(16/9)πr1^2h1] / [(1/3)πr1^2h1].
By canceling out common terms, we have V2/V1 = (16/9).
Therefore, when the radius of a cone is quadrupled and the height is reduced to 1/3 of its original size, the new volume is 16/9 times the original volume.
oldv = 1/3 pi r^2 h
Now replace r with 4r and h with whatever fraction of h, say, 1/n, it has become.
new v = 1/3 pi (4r)^2(1/n * h)
= 1/3 pi * 16r^2 * h/n
= oldv * 16/n