Final answer:
To find the radius of the cone's base, we can use the formula for the volume of a cone and solve for the radius. The expression that represents the radius of the cone's base is 4√(6x^2) / √π.
Step-by-step explanation:
In order to find the radius of the cone's base, we need to use the formula for the volume of a cone, which is V = 1/3 * π * r^2 * h, where V is the volume, π is pi, r is the radius of the base, and h is the height. We are given that the volume of the cone is 32x^3 cubic units and the height is x units. Substituting these values into the formula, we get 32x^3 = 1/3 * π * r^2 * x. To isolate the radius, we can divide both sides of the equation by 1/3 * π * x, which gives us r^2 = (32x^3) / (1/3 * π * x). Simplifying further, we get r^2 = 96x^2 / π. Taking the square root of both sides, we find r = √(96x^2 / π), which simplifies to r = √(96x^2) / √π. Since we are asked to represent the radius in units, we can simplify further to r = 4√(6x^2) / √π. Therefore, the expression that represents the radius of the cone's base is 4√(6x^2) / √π.