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)
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.
Please I need a response, I was given an assignment at school today
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
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
To find the depth of the water in the cone, we can use the formula for the volume of a frustum of a cone. The formula is:
V = (1/3) * π * h * (r^2 + R^2 + r * R)
Where:
V = volume of the frustum
h = height of the frustum
r = smaller radius of the frustum (radius of the water level)
R = larger radius of the frustum (radius of the base of the cone)
We are given the diameter of the cone, which is 12cm. We can calculate the radius of the cone:
r = (diameter/2) = 12/2 = 6cm
The larger radius of the cone is equal to the radius of the base, so R = 6cm.
Now, we can substitute the given values into the formula and solve for h:
V = (1/3) * π * h * (6^2 + 6^2 + 6 * 6)
V = (1/3) * π * h * (36 + 36 + 36)
V = (1/3) * π * h * 108
V = (36/3) * π * h
V = 12πh
Since the cone is exactly half full, the volume of water in the cone is half the volume of the entire cone. Therefore, V = (1/2) * (1/3) * π * 6^2 * 16:
(1/2) * (1/3) * π * 6^2 * 16 = 12πh
Now we can solve for h:
h = [(1/2) * (1/3) * π * 6^2 * 16] / (12π)
h = [1/6 * 36 * 16] / 12
h = 96 / 12
h = 8
Therefore, the depth of the water in the cone is 8cm.