Evaluating Volume of Curved Wedge Cut from Cylinder

[PhilthyPhil]

Trying to do this question: 'Evaluate the volume of the following solids... A curved wedge that is cut from a cylinder of radius 3m by two planes. One plane is perpendicular to the axis of the cylinder. The other plane crosses the first plane at 45 degree angle at the centre of the cylinder.'

My problem is that I am not sure what the question is asking... does it just mean a sector? In which case it would just be 9/8*Pi*length or is it something else?

 

[Galileo]

I think it's the volume enclosed by the cylinder and the two planes. There are two encosed volumes, but they have equal shapes.

I'd take the cylinder $x^2+y^2=r^2$ and the xy-plane. You pick the other plane. And draw a picture ofcourse.

 

[quicknote]

Thank you for reaching out for clarification on this question. From the description provided, it seems that the question is asking for the volume of a specific shape, which is a curved wedge that is cut from a cylinder. This shape is not a traditional sector, as it is cut at an angle and has curved sides.

To calculate the volume of this shape, we can use the formula for the volume of a wedge, which is (1/6)*pi*r^2*h, where r is the radius of the circular base and h is the height of the wedge. In this case, the radius of the cylinder is given as 3m, so we can substitute that into the formula.

However, we also need to consider the angle at which the wedge is cut. The first plane is perpendicular to the axis of the cylinder, which means it cuts the cylinder into two equal halves. The second plane intersects the first plane at a 45 degree angle at the center of the cylinder. This creates a triangular cross-section for the wedge.

To find the height of the wedge, we can use the Pythagorean theorem, where the hypotenuse is the radius of the cylinder (3m) and one of the legs is the height of the wedge (h). The other leg can be found by dividing the radius (3m) by 2, since the wedge is cut into two equal halves by the first plane. This gives us the equation h^2 + (3/2)^2 = 3^2. Solving for h, we get h = √(9 - 9/4) = √(27/4) = √(27)/2 = 3√(3)/2.

Now, we can substitute this value for h into the formula for the volume of a wedge, along with the given radius of 3m. This gives us a final volume of (1/6)*pi*3^2*3√(3)/2 = (1/6)*pi*9*3√(3)/2 = 9/4*pi*√(3) cubic meters.

I hope this helps clarify the question and provides a solution to the problem. If you have any further questions or need further clarification, please don't hesitate to ask. Thank you.