Question
A cone is exactly half full of water by volume.How deep is the water in the cone? With diameter 12cm and height 16cm.(Applying frustum of a cone)
Answers
Please I need a response, I was given an assignment at school today
i think its 8 cm (half full)
Hello Ben, it's nice hearing about your opinion.But I am sorry,the answer is 12.7,BUT I AM LOST ON HOW TO SOLVE IT
if the diameter is 12, then the radius is 6
since the height is 16, for a height h of water, r/h = 6/16 = 3/8
so, if the cone is half full,
1/3 πr^2 h = 1/2 * 1/3 π*6^2*16
(3/8 h)^2 h = 288
h^3 = 576 * 64/9 = 2048
h = 8∛4 = 12.7
If the water is only 8 cm deep, then due to similarity, its volume is only 1/8 that of the full cone.
since the height is 16, for a height h of water, r/h = 6/16 = 3/8
so, if the cone is half full,
1/3 πr^2 h = 1/2 * 1/3 π*6^2*16
(3/8 h)^2 h = 288
h^3 = 576 * 64/9 = 2048
h = 8∛4 = 12.7
If the water is only 8 cm deep, then due to similarity, its volume is only 1/8 that of the full cone.
volume of full cone = (1/3)π(6^2)(16) cm^3
= 192π cm^3
Now what height and what radius do we need to have a
volume of 96π cm^3 ?
but we also know that r/h in the cone is 6/16 or 3/8
so r/h = 3/8
r = 3h/8
(1/3) π (3h/8)^2 (h) = 96π
(3/64)π h^3 = 96π
h^3 = 2048
h = 12.699 cm
= 192π cm^3
Now what height and what radius do we need to have a
volume of 96π cm^3 ?
but we also know that r/h in the cone is 6/16 or 3/8
so r/h = 3/8
r = 3h/8
(1/3) π (3h/8)^2 (h) = 96π
(3/64)π h^3 = 96π
h^3 = 2048
h = 12.699 cm