A piece of wood whose ends are square was cut from a cylindrical piece of wood which was cut from the same pieces of wood whose ends are squares 10 centimeter by 10 centimeter and length 20 centimeter. Find the volume of wood(smallest volume) with the least waste.
To find the volume of wood with the least waste, we need to maximize the volume of the cylindrical piece that can be cut from the given piece of wood.
Let's start by calculating the maximum possible volume of a cylinder that can be cut from the given piece of wood.
The piece of wood has ends that are squares with sides measuring 10 centimeters. The area of one square end is 10 cm * 10 cm = 100 cm^2.
Since both ends are squares, the total area of both ends is 2 * 100 cm^2 = 200 cm^2.
The length of the piece of wood is 20 centimeters, so the lateral surface area of the cylindrical piece is the circumference of the end multiplied by the length. The circumference of a circle with a diameter of 10 centimeters is π * 10 cm = 10π cm.
The lateral surface area is 10π cm * 20 cm = 200π cm^2.
Now, the volume of the cylindrical piece can be calculated by multiplying the base area (the area of one end) by the height (length).
Volume of the cylindrical piece = base area * height
= 100 cm^2 * 20 cm
= 2000 cm^3
Therefore, the volume of wood with the least waste is 2000 cm^3.
To find the volume of wood with the smallest waste, we need to consider the dimensions and cuts made.
1. Start by finding the volume of the original cylindrical piece of wood.
- The base of the cylinder is a square with side length 10 centimeters.
- The height of the cylinder is 20 centimeters.
- The formula to calculate the volume of a cylinder is V = πr²h, where r is the radius and h is the height.
- Since the base is a square, the radius is half the length of a side, so r = 10/2 = 5 centimeters.
- Therefore, the volume of the cylindrical piece of wood is V = π(5)²(20) = 500π cubic centimeters.
2. Next, consider the piece of wood that was cut from the cylindrical piece.
- The ends of this piece of wood are square, so they have a side length of 10 centimeters (same as the original base).
- The height of this piece will be some value less than 20 centimeters to ensure the least waste.
- Let's assume this height as h centimeters.
3. To minimize waste, we should make sure the cut is done in such a way that the height of the remaining piece (after the square ends are cut) is minimized.
- Since the ends of the remaining piece are square, the height should also be h centimeters.
4. Now, let's calculate the volume of the remaining piece of wood.
- The base of the remaining piece is a square with side length 10 centimeters.
- The height of the remaining piece is h centimeters.
- Therefore, the volume of the remaining piece is V' = 10²h = 100h cubic centimeters.
5. We want to minimize the waste, which is the difference between the original volume and the remaining piece's volume.
- So, the waste is W = V - V' = 500π - 100h.
To minimize the waste, we need to find the value of h that minimizes the expression 500π - 100h. However, we need more information to determine the exact value of h or to compare the waste for different values of h.