Volume of 4 intersecting cylinders

[Alex3535]

Hello.Can you please help me find an expression for the volume of the left body which is actually 4 cylinders with diameter = D placed on the vertices of the cube with side length L (the cylinders are being "cut" by the corners of the cube). Thank you!

 

[jvicens]

Hello there! Thank you for reaching out for help with your question. I am happy to assist you in finding an expression for the volume of the left body that you described.

To start, let's break down the problem into smaller parts. We have a cube with side length L and 4 cylinders with a diameter of D placed on the vertices of the cube. We can visualize this as a cube with 4 smaller cubes attached to each of its corners. Each of these smaller cubes will have a cylinder "cut" through its center, creating a hollow space inside.

To find the volume of the left body, we need to find the volume of each of these smaller cubes and subtract the volume of the cylinder that has been "cut" through it. Let's call the volume of the left body V, the volume of each smaller cube Vc, and the volume of the cylinder Vcy.

The volume of a cube is given by V = L^3, where L is the side length. Since we have 4 smaller cubes, the total volume of all 4 cubes would be 4Vc. Similarly, the volume of a cylinder is V = πr^2h, where r is the radius and h is the height. In this case, the radius would be half of the diameter, so r = D/2. The height of the cylinder would be the same as the side length of the cube, so h = L. Therefore, the total volume of all 4 cylinders would be 4Vcy.

To find the volume of the left body, we can use the following expression:

V = 4Vc - 4Vcy

Substituting the expressions for Vc and Vcy, we get:

V = 4(L^3) - 4(π(D/2)^2L)

Simplifying this further, we get:

V = 4L^3 - πD^2L

Therefore, the expression for the volume of the left body is:

V = 4L^3 - πD^2L

I hope this helps you in finding the volume of the left body in your problem. Let me know if you have any further questions. Best of luck!