A new volcano has suddenly appeared where a small strip mall used to be. You are the head of the intrepid surveying crew whose job it is to determine the height of the peak. Traveling bravely toward the volcano, you stop at a safe distance and measure an angle of elevation of 21� to the top. You continue resolutely for another half mile and measure an angle of elevation of 35�. You and your crew had better not get any closer just now — there’s a new spew of ash coming out from the peak. So retreat, do some calculating, and tell the waiting reporters just how tall the volcano is. (They’ll want the answer in feet, of course.)

## Set 12

## Crumbs on a Table

A maid was about to wipe the crumbs off a rectangular table measuring 4 ft. by 8 ft. when the butler, a methodical type, took over. “You need a system!” he said. They both had very short arms and could only reach 2 feet with the dust cloth, so the butler decided to get all the crumbs together at center point C along one side. He walked along the opposite side, AB, pushing all the crumbs as far as he could directly toward C, and then did the same thing walking down each end; finally he went to the other long side to pull everything toward the dustpan. When he finished, the maid complained. Why?

## Perimeter Ratio, Squares

Given a circle of radius *x*, a square inscribed in the circle, and another square circumscribed about the circle, what is the ratio of the perimeter of the outer square to that of the inner square?

## Find Lengths of Two Sides

In △ABC, the measure of ∠A is 60°, the measure of ∠B is 45°, and the length of side AC is 8. What are the lengths of the other two sides? Do not use the Law of Sines! (You may use it afterwards to check.)

## Width of Alley

In a narrow, sinister alley of width *w* a ladder of length *a* is placed with its foot at point P between the walls. Resting against one wall at Q, a distance *k* above the ground, the ladder makes a 45° angle with the ground. Resting against the other wall at R, a distance *h* above the ground, the ladder makes a 75° angle with the ground. The width of the alley is equal to

*a*- RQ
*k*- (
*h*+*k*)/2 *h*

## Show Angle is 30°

△ABC and CA = CB, ∠C = 20°.

Also, BD = DC and BE = BA.

Show that ∠BDE = 30°.

Hint: With center B and radius BD, construct an arc of a circle meeting BA (extended) at X and BC at Y. Draw the segments DX, DY; ponder.

## Unusual Sequence

Here is a rather unusual sequence: 125, 59, 42, 34, 14, 4, ____. Can you finish the sequence?

## Defined Operation 2; 5 Parts

The operation *x*,*y* is defined by *x*,*y* = (*x* + 1)(*y* + 1) – 1 = *xy* + *x* + *y*. Which of these statements is false?

*x*,*y*=*y*,*x*for all real*x*and*y*.*x*,(*y*+*z*) = (*x*,*y*) + (*x*,*z*) for all real*x*,*y*, and*z*.- (
*x*– 1),(*x*+ 1) = (*x*,*x*) – 1 for all real*x*and*y*. *x*,0 =*x*for all real*x*.*x*,(*y*,*z*) = (*x*,*y*),*z*for all real*x*,*y*, and*z*.

## Ages as Perfect Square

Gil is 17 years older than Sheila. If his age is written after hers, the result is a 4-digit perfect square. The same statement will also be true 13 years from now. Find Sheila’s present age.

## Barbells Sale

Jasper sold one set of barbells for 99 euros (he’s Dutch, remember), and he made a 10% profit on the deal. He sold a second set for 99 euros, but this time he took a 10% loss. Counting the two sales together, did he gain, lose, or come out even?