Sunday, June 04, 2017

Geometry Problems of the Day

The following problems were taken from the GEOMETRY (COMMON CORE) Regents Exam given on Thursday, January 26, 2017.
Previous problems can be found here.

Part 1

23. A plane intersects a hexagonal prism. The plane is perpendicular to the base of the prism. Which two-dimensional figure is the cross section of the plane intersecting the prism?

(4) rectangle.
A hexagonal prism has a six-sided hexagon on its "bottom" and "top". Imagine a hexagonal building. The walls holding up the roof would be shaped like rectangles, going straight up, perpendicular to the ground.
Each of these walls part of planes that would be perpendicular to the base. So the answer is rectangle.

24. A water cup in the shape of a cone has a height of 4 inches and a maximum diameter of 3 inches. What is the volume of the water in the cup, to the nearest tenth of a cubic inch, when the cup is filled to half its height?

(1) 1.2
The equation for Volume of a cone is V = 1/3 π r2h, however, in this case, we only want 1/2 of the height. There are TWO problems with the radius. First, we're given the diameter of the top of the cone, not the radius. The radius of the top of the cone is 1.5, not 3. However, that's NOT the radius that we want. We need the radius of the circle that is halfway down the cone.
Luckily, the smaller cone and the larger cone are similar (have the same shape), so the radius is proportional. At half the height, the radius is also half, or 0.75.
Plug in these values and you have V = 1/3 (3.141592...) (0.75)2 (2) = 1.178097..., which rounds to 1.2.

Did you get tripped up by that one?

That's the end of Part I. I hope you all did well.

Continue to the next problems.


Anonymous said...

The last question is confusing because it asks for 1/2 the height... Why isn't the radius 1.5?

(x, why?) said...

I wouldn't want to say that it's meant to confuse you, but it does. I would have preferred that it were a Part II or III question for the steps involved.

Think of an ice cream cone, or a clown hat, or any other cone that comes to mind.

Think of the distance around the opening at the end of the cone if you were to run your finger around it.

Now think about tracing your finger around the outside of the cone halfway down. It's much smaller. The question is: How much smaller is it?

If you think of the diameter of that "middle" circle as a midsegment, it is half of the length of the diameter of the full cone.

So the diameter at the top is 3. The diameter in the middle is 1.5. The radius in the middle is 0.75.

Make sense?

Anonymous said...

Makes much more sense :)

Anonymous said...

I have another question-on the regents will a cross section question tell you what direction (vertical or horizontal) the shape is being crossed or cut???

(x, why?) said...

A cross section can go in any direction, even at an angle

A plane through a cylinder could be a circle, an ellipse (oval) or even a rectangle