Quiz Question

One of a continuing series

mathematicscomputeangle

Mathematics again. What is the value of the indicated angle? Post your answer as a comment below.

Update

Three people have submitted correct answers, all holding Ph.D. degrees, but none in mathematics. Here is my solution, which I believe to be the simplest approach. See the diagram:

mathematicscomputeangle-02

It’s the same as the original diagram, but I have added some labels, and I have added line BC.

Notice immediately that BC is the same as AB. If you don’t notice this immediately, then stop reading now and get into another line of work. Now notice that angle BAD is the same as angle EBC. Again, if you don’t notice, stop reading. BAD = EBC implies ABC is a right angle. Again, you can quit while you’re ahead. We have a right isosceles triangle, which means that BAC is 45°. And no mental gymnastics have been required.

Quiz Question

One of a continuing series

math-angleproblem

This popped up on my Facebook time line, posted by somebody else. So I stole it, and here it is: What is the sum of all the blue angles? Post your answer as a comment below.

Update

A number of people have posted responses, so I am going to supply the answer. See the following diagram:

math-polygonangles

What is the sum of interior angles of a polygon? The example of a triangle explains. The triangle is ABC, defined by its three interior angles. But concentrate on the complementary angles a and b and c. What is the sum of those angles? Consider the line ab. Line bc branches off from ab with a change of direction equal to angle b. Follow the path around the triangle, and the total change of direction is 360 degrees. That’s going to be the total of a and b and c. The sum of a and A is 180° so the sum of all angles is 3 × 180 = 540. 540 – 360 = 180, the sum of interior angles of all triangles.

The method holds true for all polygons. The polygon in this puzzle is unusual in that the path makes two complete turns or 360 × 2 = 720. There are 6 sides and six interior angles, so the sum of the interior angles is 6 × 180 – 720 = 360.

Quiz Question

One of a continuing series

img_3100

This is going to be very easy for most. Some will trip up on it.

There are three closed boxes. One box has only apples. One has only bananas. One has a mixture of apples and bananas. Each box is labeled to identify the contents. The problem is, all the labels are wrong. You are allowed to peek inside one box and are required to determine the contents of the two remaining boxes. Which box do you open and look into?

Stop

If you have gotten to this point you have gone too far. You should not have to think about this problem to solve it. Just post your answer in the comments section below. And also tell everybody why this didn’t require working through the possibilities to come up with the correct answer.

Quiz Question

One of a continuing series

mathematics-numbers-01

This week’s Quiz Question should be easy. Easy for those who stayed awake in class. As I understood it, this was explained once and then never again.

Myriad is a number. What number? How much is a myriad?

You can look this up on the Internet, but don’t. Enter your answer as a comment below.

Update

Greg wins. His is the correct answer. A myriad is 10,000. How this name came about, I do not know. It’s time to look it up:

myriad (n.) 1550s, from Middle French myriade and directly from Late Latin myrias (genitive myriadis) “ten thousand,” from Greek myrias (genitive myriados) “a number of ten thousand, countless numbers,” from myrios (plural myrioi) “innumerable, countless, infinite; boundless,” as a definite number, “ten thousand”

Greg was paying attention in grade school.

Quiz Question

One of a continuing series

mathematics-numbers-01

I stole this from somebody else:

Continue the following number series with the group of numbers below which best continues the series?

1 10 3 9 5 8 7 7 9 6 ? ?

11 5
10 5
10 4
11 6

Provide your answer in the comments section.

Update and Answer

Take a look at the sequence. The first number and every second thereafter increases by 1. The second number, and every second thereafter decreases by two. Therefore the next two number in the sequence are 10 and 4.

Update and Correction

Actually, I got it backward. It’s the odd numbers that increase by 2, and it’s the even numbers that decrease by 1.

Quiz Question

One of a continuing series

Math-CircleRectangle

 

The diagram shows a circle of diameter D and a rectangle. Without using trigonometry, what is the length of the line A?

Post your answer as a comment below.

Update and solution

Steve, who has some experience with mathematics, was the first to solve this one. Obviously this was a trick question, because the answer should have been staring readers in the face. See the diagram. No additional explanation will be provided.

Math-CircleRectangleSolution

The inspiration for this Quiz Question came from another Martin Gardner Book of Mathematical Problems and Diversions. This is the first edition. Last week’s Quiz Question came from the second edition. The problem is on page 111.

The idea is to make the problem appear hard when it is not. Apparently I am not able to fool all the people all the time.

Quiz Question

One of a continuing series

Mathematics-CorkPlug

I obtained this one from a book. Above is the illustration. There is a plug carved out of cork so that it can fit in each of the triangular, circular, and square holes. The bottom surface is a circle, and the top edge is a straight line, equal to the the diameter of the circle and parallel to the plane of the circle. The curved surface is generated by lines reaching from the circumference of the circle to the upper line. The lines lie in parallel planes, which in turn are perpendicular to the circle. Of course, the illustration shows only a subset of the generating lines. Hopefully by now you have the picture.

This week’s Quiz Question is: what is the volume of the plug in terms of the diameter of the circle? I did not look at the answer in the book, but I am sure I have figured this one out. The book says it’s not necessary to use calculus to compute the volume. My method uses only methods of solid geometry, which course I took in high school.

Post your answer as a comment below.

Update and solution

I am first going to post my solution. Then I’m going to the book and have a look at their solution. Some explanation:

The statement of the problem in the book says that you don’t need calculus to work this problem. If you take a look at it your first temptation might be to integrate over the enclosed shape and derive the volume. That would be using integral calculus. You would evaluate an integral such as this one:

Math-CorkPlugIntegral

That’s the something I will avoid. My approach to avoid using anything beyond simple math is to visualize the problem. Do this. Cut a slice through the cork plug from top to bottom, perpendicular to the top ridge. The cross-section looks like this:

Math-CorkPlugSolution

The triangle is the cross-section of the cork plug at some arbitrary place through the top ridge. The rectangle is the cross-section of a cylinder the same height and diameter as the plug. The triangle has ½ the area of the rectangle.

The volume of the cork plug is the sum of many thin triangular slices, such as those shown. The volume of the cylinder is the sum of many rectangular slices, such as those shown. The volume of the cork plug is ½ the volume of the cylinder.

And now for the answer from the book. The book is Martin Gardner’s Mathematical Puzzles & Diversions, from 1959. The solution is given on page 58:

4. Any vertical cross section of the cork plug at right angles to the top edge and perpendicular to the base will be a triangle. If the cork were a cylinder of the same height, corresponding cross sections would be rectangles. Each triangular cross section is obviously 1/2 the area of the corresponding rectangular cross section. Since all the triangular sections combine to make up the cylinder, the plug must be 1/2 the volume of the cylinder. The cylinder’s volume is 2π, so our answer is simply π. (This solution is given in “No Calculus, Please,” by J. H. Butchart and Leo Moser in Scripta Mathematica, September-December 1952.)

Actually, the process used to compute this volume is what formed the basis for Newton’s integral calculus. However, since no math formula is involved, we get away with saying we didn’t use calculus.

This week’s Quiz Question will be posted tomorrow at 5 a.m. It might be a bit harder. Then maybe not.