poorly worded or just me?

[renegade05]

QUESTION: Find the volume of a wedge formed by a plane slicing a right-circular cylinder of radius r if the plane meets the base at an angel \(\displaystyle \theta\) and the line formed by its intersection with the base forms a diameter of the cylinder.

Ok, I dunno if i have just been doing too much math lately and my brain has quit on me, or this wording is very awkward.

Nevertheless, I am not sure what this question is asking. It is a practice problem given for an upcoming test. Can someone maybe reword it, or explain to me what I am suppose to do.

Thanks.

P.S. I even have the solution but the picture would honestly give a hieroglyphologist a hard time!

 

[galactus]

I think I know what they mean. I am a amateur Egyptologist. Picture cutting a wedge from a can.

If the x-axis is along the diameter of the cylinder, then the base is a semicircle of radius r with equation \(\displaystyle y=\sqrt{r^{2}-x^{2}}\).

A cross section perp. to the x-axis at a distance x from the origin is a right triangle. This triangle has base \(\displaystyle y=\sqrt{r^{2}-x^{2}}\)

The height of this triangle is \(\displaystyle ytan(\theta)\Rightarrow \sqrt{r^{2}-x^{2}}tan(\theta)\)

A triangle has area \(\displaystyle \frac{1}{2}bh\).

So, the area of the cross section is \(\displaystyle A=\frac{1}{2}\underbrace{\sqrt{r^{2}-x^{2}}}_{\text{base}}\cdot \underbrace{tan(\theta)\sqrt{r^{2}-x^{2}}}_{\text{height}}=\frac{tan(\theta)}{2}(r^{2}-x^{2})\),

So, as those triangles 'stack up' along the semicircle, we add up their areas. To do this, we integrate.

Integrate to find the volume of said wedge: \(\displaystyle \frac{1}{2}tan(\theta)\int_{-r}^{r}(r^{2}-x^{2})dx\)

I think this is what they're referring to. Note the semicircle in the base when the plane cuts through the diameter of the base.

The bottom of a cylinder is a circle. If you cut it in half you have a semicircle.

Now, if you have a known angle and radius, you can find the volume of the wedge because this is a general formula.

Let me know what you get.

 

[soroban]

Hello, renegade05!

I had to think about it . . . but I think I understand.

Find the volume of a wedge formed by a plane slicing a right-circular cylinder of radius \(\displaystyle r\)
if the plane meets the base at an angle \(\displaystyle \theta\)
and the line formed by its intersection with the base forms a diameter of the cylinder.

The last sentence means: the plane cuts through the center of the circular base.

      *-----------*
      |           | /
      |           |/
      |           /
      |          /|
      |         / |
      |        /  |
      |       /   |
      |      /@   |
      *-----*-----*
         r /   r
          /

Edit: Too slow again . . . galactus beat me to it.
.