Math Problems, Number Theory Problems

Problem of the Day: 1/15/13

How many four-digit numbers are divisible by 11 and have a sum of digits of 10?

Solution to yesterday’s problem:

Let’s designate the square’s side length a. Then the area of the square is a^2. To find the area of the triangle, we need to find its height. First we’ll draw the altitude to make two 30-60-90 right triangles (whose side lengths are s, s\sqrt{3}, and 2s). The short side of each triangle is \frac{1}{2}a, so the vertical side is \frac{1}{2}a\sqrt{3}. Using the triangle area formula,

A=\frac{1}{2}a(\frac{1}{2}a\sqrt{3})

A=\frac{1}{4}a^2\sqrt{3}

Now we can write our two areas as a ratio:

\frac{\frac{1}{4}a^2\sqrt{3}}{a^2}

Canceling and multiplying both sides by four, we get the final answer \frac{\sqrt{3}}{4}.

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s