Find the volume of the space between the cylinder and the cone. Note the cone is touching the top of the cylinder.
The height of the cylinder is
11
11 inches and the diameter is
48
48 inches.
Use 3.14 for π. Round your answer to the nearest tenth
First, we need to find the radius of the cylinder by dividing the diameter by 2:
Radius = 48 / 2 = 24 inches
Next, we find the volume of the cylinder using the formula for the volume of a cylinder:
Volume of cylinder = π * radius^2 * height
Volume of cylinder = 3.14 * 24^2 * 11
Volume of cylinder = 3.14 * 576 * 11
Volume of cylinder = 20045.28 cubic inches
Now, let's find the volume of the cone. Since the cone is touching the top of the cylinder, the height of the cone will be equal to the height of the cylinder.
The radius of the cone will be the same as the radius of the cylinder, which is 24 inches.
Using the formula for the volume of a cone:
Volume of cone = 1/3 * π * radius^2 * height
Volume of cone = 1/3 * 3.14 * 24^2 * 11
Volume of cone = 1/3 * 3.14 * 576 * 11
Volume of cone = 6047.52 cubic inches
Now, we find the volume of the space between the cylinder and the cone by subtracting the volume of the cone from the volume of the cylinder:
Volume = Volume of cylinder - Volume of cone
Volume = 20045.28 - 6047.52
Volume = 14097.76 cubic inches
Therefore, the volume of the space between the cylinder and the cone is approximately 14097.8 cubic inches.
wrong
I apologize for the error in my previous calculation. Let's correct it.
To find the volume of the space between the cylinder and the cone, we need to consider that the cone is touching the top of the cylinder. This will form a frustum, which is the part remaining after cutting off the top of a cone by a plane parallel to its base.
Given:
- Height of the cylinder (h): 11 inches
- Diameter of the cylinder (d): 48 inches
- Radius of the cylinder (r): d / 2 = 24 inches
The slant height (l) of the cone can be found using the Pythagorean theorem:
l^2 = r^2 + h^2
l^2 = 24^2 + 11^2
l^2 = 576 + 121
l^2 = 697
l = √697
l ≈ 26.4 inches
Now, to find the radius (R) of the cone at its base, we can use similar triangles:
r / R = h / (l - h)
24 / R = 11 / (26.4 - 11)
24 / R = 11 / 15.4
R = 24 * 15.4 / 11
R ≈ 33.6 inches
Now we can calculate the volume of the frustum (the space between the cone and the cylinder) using the formula:
V = 1/3 * π * h * (r^2 + rR + R^2)
V = 1/3 * 3.14 * 11 * (24^2 + 24*33.6 + 33.6^2)
V = 1/3 * 3.14 * 11 * (576 + 806.4 + 1128.96)
V = 1/3 * 3.14 * 11 * 2511.36
V ≈ 29292.06 cubic inches
Therefore, the volume of the space between the cylinder and the cone is approximately 29292.1 cubic inches.