Question

A plane inclined at an angle of 45 passes through a diameter of the base of a cylinder of radius (r). Write an expression for the volume (V) of the region within the cylinder and below the plane in terms of (r). (Express numbers in exact form. Use symbolic notation and fractions where needed.)

a) (V(r) = 1/2 π r^3)
b) (V(r) = π r^2)
c) (V(r) = 1/2 π r^2)
d) (V(r) = π r^3)

Answer 1

The Correct option (a) The expression for the volume (V) of the region within the cylinder and below the plane in terms of (r) is V(r) = (1/2)πr³.

Step-by-step explanation:

The volume (V) of the region within the cylinder and below the inclined plane can be determined by finding the volume of the cylinder and then subtracting the volume of the region above the plane.

To find the volume of the cylinder, we use the formula:

V = πr²h

where r is the radius of the cylinder and h is the height of the cylinder.

Since the plane passes through the diameter of the cylinder's base, the height (h) of the cylinder is equal to the diameter (2r) multiplied by the sine of the angle of inclination (45 degrees).

Therefore, the volume of the cylinder is given by:

V₁ = πr²(2r*sin(45))

To find the volume of the region above the plane, we subtract the volume of the cylinder from the total volume of the cylinder:

V = V₁ - (1/2)πr²(2r*sin(45))

Once simplified, the expression for the volume (V) of the region within the cylinder and below the plane in terms of (r) is:

V(r) = πr³ - (1/2)πr³ = (1/2)πr³