Help calculus?

Question

Use method of slicing to determine the volume of the wedge cut from a right circular cylinder with radius √3 cm. by two planes, one perpendicular to the axis of the cylinder and the other intersecting the first along a diameter of the circular plane section at an angle of 60°

How will I draw the diagram? I don't know what the axis of the cylinder is... Pls help...

Answer 1

Draw a circle centre O :-

Inside the circle construct segment AOB = 60°
where A and B lie on the circumference.
Radius of cake = √ 3 cm
Let height of cake be h cm
60° / 360° = 1 / 6

Volume of slice = (1/6).π.r². h = (1/6).3π h = (π/2).h

Volume of slice can be worked out if height of cake is known.

The axis of the cylinder is the vertical line that passes thro` the centre of the circle.

Answer 2

Think of a resistor, with a wire going through the middle of a cylinder. The wire is the axis of the cylinder. In other words, it's along the length of the cylinder.

As for the picture, the first plane would have to cut the cylinder in half lengthwise, in order to intersect the other plane at the diameter of the circle. The second plane would cut an oval shape out of the cylinder, as it's slanted at 60 degrees to the first one.

Answer 3

I can picture it but I don't think I'm enough of an artist to draw it. Think of a semicircle for a base with a height that goes from zero along the diameter and increases as you move away from the diameter by an angle of 60°. So we are looking at triangular wedges along ∆y.

x² + y² = r²
x² = r² - y²

Area of a wedge A = (1/2)xz

Let the angle of the inclined plane be α. Then

tanα = z/x
z = xtanα

A = (1/2)xz = (1/2)x(xtan α) = (1/2)(x²)tanα
A = (1/2)(tanα)(r² - y²)

V = ∫A(y)dy = ∫{(1/2)(tan α)(r² - y²)}dy | [Eval on (-r,r)]
= (1/2)(tan α)(r²y - y³/3) | [Eval on (-r,r)]
= (1/2)(tan α)[(r³ - r³/3) - (-r³ + r³/3)]
= (1/2)(tan α)[4r³/3]
= [2r³/3](tan α)
= [2r³/3](tan 60°) = [2r³/3](√3) = (2/√3)r³
= (2/√3)(3√3) = 2*3 = 6

Answer 4

I can't draw this diagram here!
The axis of the cylinder is the line joining the centre of the base to the centre of the "lid" of the cylinder.