The volume of a cone varies jointly as the square of its radius and its height. If the volume of a cone is 50π cubic inches when the radius is 5 inches and the height is 6 inches, find the volume of a cone when the radius is 4 inches and the height is 3 inches
50π * (4/5)^2 * (3/6)
So would it be 16
You should know that
volume of cone = (1/3) π r^2 h
if not, then ....
V = k r^2 h
when V = 50π, r = 5, and h = 6
50π = k(25)(6)
k = 50π/(150) = π/3 or (1/3)π , how about that?
V = (1/3)π (4^2)(3) cubic inches = same answer at R_scott,
and certainly not 16
So would it be 16 π ? Ugh lol no matter what I do I keep getting 16. Thank you guys for your help trying to explain to me :)
You are told that
v = k r^2 h
That means that V/(r^2 h) = k, a constant, which does not change. So now you know you are looking for v such that
v/(4^2 * 3) = 50π/(5^2 * 6)
v/48 = 50π/150 = π/3
v = 48π/3 = 16π
To solve this problem, we need to start by writing the equation that represents the given relationship between the volume, radius, and height of a cone.
The volume of a cone can be calculated using the formula V = (1/3)πr^2h, where V is the volume, r is the radius, and h is the height.
Based on the information given, we can write a proportion to express the relationship of the volume, radius, and height:
V / r^2h = k
where k is the constant of variation.
We know that when the radius is 5 inches and the height is 6 inches, the volume is 50π cubic inches. By substituting these values into the equation, we can solve for k:
50π / (5^2 * 6) = k
Simplifying:
50π / 150 = k
k = π / 3
Now that we have the value of k, we can use it to find the volume of the cone when the radius is 4 inches and the height is 3 inches. Substitute these values into the equation:
V / (4^2 * 3) = π / 3
Simplifying further:
V / 48 = π / 3
Multiply both sides of the equation by 48 to isolate V:
V = (π / 3) * 48
V = 16π cubic inches
Therefore, the volume of the cone when the radius is 4 inches and the height is 3 inches is 16π cubic inches.