A sculpture is a design by carving a cone out of a cylinder. The cone and cylinder share the same radius and height. If the volume remaining is 144 in^3. What was the volume of the original cylinder
Let's assume the radius and height of the cylinder (and the cone carved out of it) is "r" and "h" respectively.
The volume of a cone is given by the formula V_cone = (1/3) * π * r^2 * h.
The volume of a cylinder is given by the formula V_cylinder = π * r^2 * h.
Since the cone is carved out of the cylinder, the remaining volume is given by the formula V_remaining = V_cylinder - V_cone.
Substituting the values, we get V_remaining = π * r^2 * h - (1/3) * π * r^2 * h.
Simplifying further, we get V_remaining = (2/3) * π * r^2 * h.
Given that the volume remaining is 144 in^3, we can equate the equation to 144: (2/3) * π * r^2 * h = 144.
To find the volume of the original cylinder, we need to calculate V_cylinder which is given by the formula V_cylinder = V_cone + V_remaining.
Substituting the values, we get V_cylinder = V_cone + (2/3) * π * r^2 * h.
Since the cone and cylinder share the same radius and height, we can further simplify it as V_cylinder = V_cone + (2/3) * V_cone.
Substituting the formula for V_cone, we get V_cylinder = (1/3) * π * r^2 * h + (2/3) * (1/3) * π * r^2 * h.
Simplifying further, we get V_cylinder = (1/3 + 2/9) * π * r^2 * h.
Calculating the common denominator, we get V_cylinder = (3/9 + 2/9) * π * r^2 * h.
Simplifying further, we get V_cylinder = (5/9) * π * r^2 * h.
Now, we have the volume of the original cylinder in terms of the given variables.