The volume of a right circular cone is 300 cubic inches. What is the volume, in cubic inches, of a right cylinder that has the same base and height as the cone?
a cylinder has volume Bh
a cone has volume 1/3 Bh
so its volume is 1/3 * 300 = 100
Wait, that wouldn't help because we already know that
To find the volume of the cylinder that has the same base and height as the cone, we need to determine the relationship between the volumes of the two shapes.
First, let's recall the formulas for the volume of a cone and a cylinder:
Volume of a cone: Vc = (1/3) * π * r^2 * h,
Volume of a cylinder: Vd = π * r^2 * h,
where Vc is the volume of the cone, Vd is the volume of the cylinder, r is the radius of the base, and h is the height of both the cone and cylinder.
In this case, we know that the volume of the cone is 300 cubic inches. Thus, we have:
300 = (1/3) * π * r^2 * h.
We also know that the cone and cylinder have the same base and height, so their radii and heights are equal. This means we can write r = h for both shapes.
Let's substitute r = h into the equation:
300 = (1/3) * π * (h^2) * h.
Now, let's simplify the equation:
900 = π * (h^3).
h^3 = 900 / π.
To solve for h, we need to take the cube root of both sides of the equation:
h = (900 / π)^(1/3).
Finally, we can substitute this value of h into the formula for the volume of the cylinder:
Vd = π * r^2 * h = π * (h^2) * h = π * ((900 / π)^(1/3))^2 * ((900 / π)^(1/3)).
Vd = π * (900 / π)^((2/3) + (1/3)) = π * (900 / π)^(3/3) = π * (900 / π)^(1).
Therefore, the volume of the cylinder with the same base and height as the cone is π * (900 / π)^(1) cubic inches.
The answer would be 300.
Thanks so much!
The volume of a right circular cone is 300 cubic inches.
Hope this helps!