a cone has a radius of 2 inches and a height of 12 inches if the radius of the cone is double what must be the new height in the cone in order to keep the volume constant
To find the new height of the cone when the radius is doubled while the volume remains constant, we need to first understand the relationship between the radius, height, and volume of a cone.
The volume of a cone is given by the formula V = (1/3)πr²h, where V is the volume, r is the radius, and h is the height.
Since we want to keep the volume constant, we can set up the following equation:
(1/3)π(2²)12 = (1/3)π(r²)h
Simplifying this equation, we get:
4π(12) = r²h
To find the new height, we need to double the radius. If the original radius is 2 inches, then the new radius will be 2 × 2 = 4 inches.
Substituting the new radius and the constant volume into our equation, we have:
4π(12) = (4²)h
Simplifying further:
48π = 16h
Now, to find the new height, we divide both sides of the equation by 16π:
(48π)/(16π) = h
3 = h
Therefore, to keep the volume constant when the radius is doubled, the new height of the cone must be 3 inches.