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[NevarX]

A wedge is cut from a solid in the shape of a right circular cylinder with a radius of r centimeters by a plane through a diameter of the base and inclined to the plane of the base at an angle of measurement 45°. Find the volume of the wedge.

 

[🤖]

To find the volume of the wedge, we need to calculate the volume of the original cylinder and subtract the volume of the cut-out wedge.

1. Volume of the original cylinder:
The formula for the volume of a cylinder is V = πr^2h, where r is the radius and h is the height.

In this case, the radius of the cylinder is r centimeters, and the height is also r centimeters since the plane cuts through the diameter of the base. So, the volume of the original cylinder is V1 = πr^2(r) = πr^3.

2. Volume of the cut-out wedge:
A wedge is formed by cutting out a portion of the cylinder. The wedge is inclined at an angle of 45° to the plane of the base. This means that the height of the wedge will be r/sqrt(2), as it is the hypotenuse of a right triangle with legs of length r.

The base of the wedge is a circular sector with an angle of 45°, which means it covers 45/360 = 1/8th of the total circle. The formula for the area of a circular sector is A = (θ/360)πr^2, where θ is the central angle.

So, the volume of the wedge is V2 = (1/8)(πr^2)(r/sqrt(2)) = (πr^3)/(8sqrt(2)).

3. Final calculation:
The volume of the wedge is obtained by subtracting the volume of the cut-out wedge from the volume of the original cylinder:
V = V1 - V2 = πr^3 - (πr^3)/(8sqrt(2)) = (7πr^3)/(8sqrt(2)).

Therefore, the volume of the wedge is (7πr^3)/(8sqrt(2)) cubic centimeters.