A wedge is cut from a right circular cylinder of radius r by two planes, one perpendicular to the axis of the cylinder and the other making an angle (beta) with the first. Find the volume of the wedge by slicing perpendicular to the y-axis.
Please show all steps and explainations please.
Note: this is the entire question, nothing is missing. This is the problem I got from my math teacher.
To find the volume of the wedge, we can divide it into smaller slices perpendicular to the y-axis.
First, let's visualize the situation. We have a right circular cylinder, and it's easier to work with if we consider it as a solid with circular cross-sections parallel to its base.
1. Imagine a cross-section of the cylinder, perpendicular to its axis. It will be a circle with radius r.
2. We then cut a wedge from this cross-section using two planes. The first plane is perpendicular to the axis (creating a straight edge), and the second plane makes an angle β with the first.
3. We need to find the volume of this wedge by slicing it perpendicular to the y-axis. This means that the slices we will consider should be parallel to the y-axis and have thickness dy.
To solve this problem, we will use integration.
Step 1: Determine the height of the wedge
Start by considering the cross-section of the cylinder again. The height of the wedge, h, can be determined by the angle β.
Since we want to find the volume of the wedge by slicing perpendicular to the y-axis, we will express h in terms of y.
Notice that when we slice perpendicular to the y-axis, the resulting slice will be a rectangle. The width of this rectangle is equal to the circumference of the cross-section of the cylinder, which is 2πr. The length (or height) of the rectangle is equal to h.
To find the value of h, we can use trigonometry. In the right triangle formed by the rectangle and the slanted line, the opposite side (the height of the cylinder) is r, and the adjacent side is h. Therefore, we can use the tangent function:
tan β = opposite/adjacent
tan β = r/h
By rearranging the equation, we can solve for h:
h = r/tan β
Step 2: Finding the volume of a slice
We will now find the volume of a single slice with thickness dy.
The volume of a slice can be approximated as the product of the area of the slice base and its thickness. In this case, the base of each slice will be the cross-sectional area of the wedge.
The cross-sectional area of the wedge is a rectangle with width 2πr (the circumference of the cross-section) and height dy. Therefore, the volume of the slice can be calculated as:
dV = (2πr)(dy)
Step 3: Evaluating the integral
To find the total volume of the wedge, we need to integrate the expression we found for the volume of a slice:
V = ∫[0, h] dV
V = ∫[0, h] (2πr)(dy)
Evaluate the integral:
V = 2πr * ∫[0, h] dy
V = 2πr * [y] evaluated from 0 to h
V = 2πr(h - 0)
V = 2πrh
Since we found in step 1 that h = r/tan β, we can substitute this value:
V = 2πr(r/tan β)
V = 2πr²/tan β
Therefore, the volume of the wedge, when sliced perpendicular to the y-axis, is given by:
V = 2πr²/tan β