Find the volume of a cone of height 8 centimeters and base radius 6 centimeters. This cone is sliced by a plane that is parallel to the base and 2 centimeters from it. Find the volumes of the two resulting solids. One is a cone, while the other is called a frustum.
use similar triangles (a side view) to find the radius of the slice 2cm up.
Then you can find the volume of the frustrum by subtracting the top part from the whole cone.
Why did the cone go to the therapist? Because it was feeling a little "frustum-rated" after being sliced! Alright, let's find the volume of these slices.
First, let's find the volume of the smaller cone created. The height of the smaller cone is 2 cm, and the base radius is still 6 cm. The formula for the volume of a cone is V = (1/3)πr^2h. So, the volume of the smaller cone is V = (1/3)π(6^2)(2) = 24π.
Next, let's find the volume of the frustum. The height of the frustum is 8 cm - 2 cm = 6 cm. The formula for the volume of a frustum is V = (1/3)πh(R^2 + r^2 + Rr), where R is the radius of the larger base and r is the radius of the smaller base. For the frustum, R = 6 cm and r = 6 cm - 2 cm = 4 cm. Plugging in these values, we get V = (1/3)π(6)(6^2 + 4^2 + 6(4)) = 76π.
So, the volume of the smaller cone is 24π and the volume of the frustum is 76π.
To find the volumes of the two resulting solids, we first need to calculate the dimensions of each solid.
Let's start with the smaller cone (Cone 1) that is sliced off.
Since the original cone has a height of 8 centimeters and a base radius of 6 centimeters, the smaller cone will have a height of 2 centimeters (the distance between the plane and the base) and a base radius of 6 centimeters.
The formula for the volume of a cone is given by V = (1/3) * π * r^2 * h, where V represents volume, π is a constant approximately equal to 3.14159, r is the radius, and h is the height.
For Cone 1:
V1 = (1/3) * π * (6^2) * 2
= (1/3) * 3.14159 * 36 * 2
= 75.39816 cm^3
The volume of Cone 1 is 75.39816 cm^3.
Now let's calculate the volume of the frustum (Cone 2) that remains.
The frustum is formed by the original cone (with a height of 8 cm and a base radius of 6 cm) and Cone 1 (with a height of 2 cm and a base radius of 6 cm).
To calculate the volume of the frustum, we need to subtract the volume of Cone 1 from the volume of the original cone.
For Cone 2 (frustum):
V2 = Voriginal cone - V1
= (1/3) * π * (6^2) * 8 - 75.39816
= (1/3) * 3.14159 * 36 * 8 - 75.39816
= 904.77868 cm^3 - 75.39816 cm^3
= 829.38052 cm^3
The volume of the frustum (Cone 2) is 829.38052 cm^3.
So, the volume of the smaller cone (Cone 1) is 75.39816 cm^3, and the volume of the frustum (Cone 2) is 829.38052 cm^3.
To find the volumes of the two resulting solids, we can use the formulas for the volume of a cone and the volume of a frustum.
First, let's find the volume of the smaller cone:
1. The smaller cone is formed by the slice of the original cone, with height 2 centimeters and base radius 6 centimeters.
2. The formula for the volume of a cone is V = (1/3) * π * r^2 * h, where r is the radius and h is the height.
3. Plug in the values into the formula: V = (1/3) * π * (6^2) * 2.
4. Calculate the volume: V = (1/3) * π * 36 * 2 = 24π cubic centimeters.
So, the volume of the smaller cone is 24π cubic centimeters.
Next, let's find the volume of the frustum:
1. The frustum is the remaining solid after removing the smaller cone. It has the same height as the original cone (8 centimeters), but with a smaller base radius (4 centimeters).
2. The formula for the volume of a frustum is V = (1/3) * π * (R^2 + r^2 + Rr) * h, where R is the larger base radius, r is the smaller top radius, and h is the height.
3. Plug in the values into the formula: V = (1/3) * π * (6^2 + 4^2 + 6*4) * 8.
4. Calculate the volume: V = (1/3) * π * (36 + 16 + 24) * 8 = (1/3) * π * 76 * 8 = 304π cubic centimeters.
So, the volume of the frustum is 304π cubic centimeters.
Therefore, the volumes of the two resulting solids are 24π cubic centimeters for the smaller cone and 304π cubic centimeters for the frustum.