Take the circumference of the earth at the equator to be 24,000 miles. An airplane taking off at the equator and flying west at 1,000 miles per hour would land at exactly the same time that it started, and the sun would not move in the plane’s sky during that flight. (Work this out visually in your mind.) Now, can you find a formula so that the speed of a plane flying at any particular latitude and keeping up with the sun, is a function of that latitude: speed = f(x°)?

## Set 11

## Tangent Circles, Area of Shaded Region

A circle centered at A of radius *r* is externally tangent to a circle centered at B of radius 2*r*. Tangent lines are drawn from A to the larger circle with points of tangency at D and E respectively. Find the area of the shaded portion of the diagram.

## Peanut Butter Jar

If a jar of peanut butter that is 3 inches in diameter and 4 inches tall sells for $2.00 (assuming it is cylindrical in shape), what is a fair price for a jar that is 6 inches in diameter and 6 inches tall?

## Interior Segments of Square

A square has sides each having a length of 2. Segments are drawn from one vertex to the midpoints of each of the four sides of the square. What is the sum of the lengths of these segments?

## Hard Triangle Area in Square

If GRAW is a square with side *a*, and triangle GMR is equilateral, then what is the area of triangle RAC (in terms of *a*)?

## Split Parallelogram Area

*PQRS* is a parallelogram. Let *m* be a line that passes through the parallelogram in such a way that it splits the area into two equal parts. How many such lines can there be? If you were to draw all of these lines, you would notice something about them: What is it?

## Simplify Complex Polynomial

If *w* is one of the imaginary roots of the equation *x*^{3} = 1,

then the quartic (1 – *w* + *w*^{2})(1 + *w* – w ^{2}) is equal to:

- 4
*w*^{2}- 2
- –
*w* - 1

## Giant Binomial Squared

The sum of the digits in base ten of (10^{(4n}^{2} +8) +1)^{2}, where *n* is a positive integer, is:

- 4
- 4
*n* - 2 + 2
*n* - 4
*n*^{2} *n*^{2}+*n*+ 2

## Muffins

Milly Miffin made one muffin more than Molly’s mother made, and Milly Miffin’s mother made one muffin more than Molly made; and Molly’s mother and Milly and Milly’s mother and Molly made 50 muffins (total), but Molly’s mother and Milly made 4 muffins more than Milly’s mother and Molly made. So murmur how many muffins Milly made.

## Audio Equipment

Sue got a lot of money for graduation from a mysterious, rich aunt that she’d never known she had. Her first thought was AUDIO EQUIPMENT, but then she figured she really ought to put it away for a rainy day. She divided it into two parts. One part had $250 more than the other part. She invested the smaller part at 6% interest, and the larger part at 8%. A year later, the total interest she had accumulated was $72.50. How much money did she get from her mysterious, rich aunt for graduation in the first place?