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.

Algebra 2 Problems of the Day

The following problems were taken from the ALGEBRA II (Common Core) Regents Exam given on Friday, January 27, 2017.
Previous problems can be found here

Part 1

21. Joelle has a credit card that has a 19.2% annual interest rate computations. compounded monthly. She owes a total balance of B dollars after m months. Assuming she makes no payments on her account, the table below illustrates the balance she owes after m months.

Over which interval of time is her average rate of change for the balance on her credit card account the greatest?

(4) month 60 to month 73
Look at the image below. Find the average rate of change by calculation (y₁ - y₂) / (x₁ - x₂).

22. Which graph represents a cosine function with no horizontal shift, an amplitude of 2, and a period of 2/3 π ?

(3) See below
Choices (2) and (4) are out because they start at 0 (sine graphs). Graph (1) shows a function with a period of 2/9 π, as it repeats three 3 in the space of 2/3 π. Choice (3) shows a function that repeats 3 times in the space of 2π, so it has a period of 2/3π.

Continue to the next problems

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

21. In the diagram below of circle O, GO 8 and m∠GOJ = 60°. What is the area, in terms of π, of the shaded region?

(4) 160π / 3.
Since 60° is 1/6th of the 360° degree in the complete circle, then the unshaded region of the circle is (1/6) πr² = (1/6) π8²
and the shaded portion would be (5/6) πr² = (5/6) π8² = (5 * 64π) / 6 = (5 * 32π) / 3 = 160π/ 3.

22. A circle whose center is the origin passes through the point (5,12).
Which point also lies on this circle?

(3) (11, 2 sqrt(12))
The equation of the circle is x² + y² = r². We can find r using the Distance Formula or Pythagorean Theorem: 5² + 12² = r².
25 + 144 = 169 = r²
r = 13 (which you really should have known. Look up Pythagorean triples.)

Which of the other points creates a right triangle with a hypotenuse of 13?
(10, 3) definitely do not -- they don't even create a triangle with a side of 13. (-12, 13) can't because the hypotenuse is longer than the legs (plus it would have to be -12 and either 5 or -5).
Check 11² + (2sqrt(12))² = 121 + 4(12) = 169 = 13². Looks good.
Check (-8)² + (5sqrt(21))² = 64 + 25(21) = way too much. No good.




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Bubblegum

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(C)Copyright 2017, C. Burke.

These things have a way of blowing up in your face. Kind of like bubblegum.




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Memorial Day BBQ Math

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(C)Copyright 2017, C. Burke.

So how many beers is 1 steak, 2 hot dogs and a cheeseburger and a half?




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Orbit Memory

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(C)Copyright 2017, C. Burke.

Cost a lot of Bucks ... and was gone in a Flash!

If the worksheets on the flip side of the original sketches are any indication, this pun has been waiting to happen since about 2009.

I didn't have the skills then (and, frankly, I don't really have them now, but I try more).

Whatever the scientist's name was supposed to have been is lost to time (and my own faulty Memory). Victor Vargos was the name of the scientist is a schlock satire I wrote shortly after college. He was a good guy and saved the world, but I felt I needed a name here.

And I was already days late, and switched up my comic at the late moment in the process.




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Center, Pt. 3

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(C)Copyright 2017, C. Burke.

It was almost the satyr of the centaur circle, which would have been a lot of work to throw out at the last minute!

I think I'm done beating this dead ... well, you know.

See also, Center, Pt. 1 and Center, Pt. 2.




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Another Grim Venn

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(C)Copyright 2017, C. Burke.

Perhaps this is what you were expecting the last time?

Cross reference: A Grim Venn




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Village Squares

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(C)Copyright 2017, C. Burke.

"If you know just two sides
There's a rule that provides
How to find the space confined.
If the angle that's included
Has a value that computed
You can use the Law of Sines!"




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A Grim Venn

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(C)Copyright 2017, C. Burke.

Yes, it's grim, but it's also fantastic!




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Center, Pt. 2

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(C)Copyright 2017, C. Burke.

Yes, I've said this in class. As any fun teacher would. Amirite?

And for the geeks out there: May the Fourth be with you. I didn't forget, but I didn't have time to make a special comic. I do Star Wars stuff throughout the year when the mood strikes.




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Center, Pt. 1

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(C)Copyright 2017, C. Burke.

Nice to know things are fine.

Warning you now, that there's a part two because I couldn't decide on which joke to use, so I'm using them both.
And I won't even try to hide it by doing the other one two or three weeks from now.




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