Calculate the height of a cylinder of maximum volume that can be cut from a cone of height 20 cm and base radius 80 cm.
once is enough
To find the height of a cylinder of maximum volume that can be cut from a cone, we can use calculus.
Let's denote the height of the cylinder as h and its volume as V. The radius of the cylinder will be equal to the base radius of the cone.
The volume of a cylinder is given by the formula: V = πr^2h, and the volume of a cone is given by the formula: V = (1/3)πr^2h.
Since we want to maximize the volume of the cylinder, we need to optimize the volume function V with respect to the height h.
To do this, we can set up the following equation by equating the volume of the cylinder with the volume of the cone:
π(80)^2h = (1/3)π(80)^2(20-h).
Simplifying the equation, we get:
6400h = 1/3 * 6400 * (20 - h).
Now, we can solve this equation for h:
6400h = (1/3)(6400)(20 - h).
Multiplying through by 3:
3 * 6400h = (6400)(20 - h)
19200h = 128000 - 6400h
19200h + 6400h = 128000
25600h = 128000
Dividing both sides by 25600:
h = 128000 / 25600
h = 5
Therefore, the height of the cylinder of maximum volume that can be cut from the given cone is 5 cm.