Final answer:
The volume of a cone varies jointly as its height and the square of its radius. By plugging the given values into the equation and solving for the constant, we can find the volume of a cone with different height and radius.
Step-by-step explanation:
The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height of the cone.
In this problem, we are told that the volume of a cone varies jointly as its height and the square of its radius. This means that the volume (V) is equal to a constant (k) times the product of the height (h) and the square of the radius (r^2), or V = khr^2.
We are also given that a cone with a height of 8 cm and a radius of 2 cm has a volume of 33.5 cm. So we can plug those values into the equation and solve for the constant k: 33.5 = k(8)(2^2)
Simplifying the equation, we get: 33.5 = 32k
Dividing both sides by 32, we find that k = 1.046875
Now, we can use this value of k to find the volume of a cone with a height of 6 cm and a radius of 4 cm: V = 1.046875(6)(4^2) = 100.125 cm