Aussie Lake is possibly the most amazing lake in the world. This is because it was formed into a perfect circle of water. A sailor sailed directly across the middle and found that it was 800 yards across. There is a walkway 74 feet outside the perimeter of the lake that goes completely around it. Todd and Joe are planning on biking along the pathway but they need to know how far they would have to bike.

To solve this we first converted yards into feet and found that the diameter (or the distance across that the sailor found) was 800 X 3 feet. We could now determine that Aussie Lake�s diameter is 2,400 feet. However, we then realized that we needed to find the diameter of the pathway. We added 148 feet to the diameter of the lake because 148 is the quantity of 2 times 74 feet, which was how many feet the pathway was from the lake. The diameter of the walkway is therefore 2,548 feet. We knew that the circumference of the walkway was diameter(pi). We multiplied 2548 by pi. Finally, we could determine that we would have to bike 8,004.78 feet to go around the lake using the pathway

Another amazing fact about Aussie Lake is that its depth is 92 feet deep everywhere. We have heard about a hidden treasure that lies directly in the center of the lake (lying on the bottom.) We want to be the first to get to the treasure before everyone else does. Therefore, we want to find the shortest distance to the treasure (from the edge of the shore directly to the treasure.) How far is the shortest distance to the treasure?

To find this measure, we had to realize that the distance is merely a hypotenuse of a right triangle. Side 1 is the radius of the lake (800 / 2 = 400) which is 400 feet. Side 2 is the depth of the lake, which is 92 feet. To find the shortest distance to the treasure we used the Pythagorem Theorem. 400 squared + 92 squared = x squared. We did the math and found that it is 410.44 feet to the treasure and that we would have enough oxygen in our air tanks to get there and back.

Our math knowledge paid off and we received the treasure box. When we returned, we could not open the box. It was a cylindrical box that had the inscription: �SA Is the key.� There was also a numerical keypad. We thought about this strange message and finally came to the hypothesis that SA stood for surface area and by typing in the correct surface area of the box, we could unlock the treasure. How could we find the surface area?

We first measured the diameter of one of the circular ends that was 44 inches. We could then find the area of the two circular ends. Each end was pi(r)squared. The radius was 22 inches and the area was therefore 484(pi) = 1,520.53 sq. inches. Since there were two circular ends, we multiplied that number by two and got 3,041.06 sq. inches for the two ends. There was still the task of getting the area of the main part of the cylindrical box. We realized that, if unfolded, the main part would merely be a rectangle with the circumference of the circular ends as the width and the height of the box as its height. We had to find the circumference of the circular end. We knew that it was the diameter (44 inches) times pi. We got 138.23 inches as the circumference. We then measured the height of the box and found that it was 50 inches. We multiplied 50 x 138.23 and got 6,911.5 inches. Finally, we added 6,911.5 to 3,041.06 and determined that the surface area of the box was 9,952.56 inches squared.

We typed in the surface area we received and instantly the box opened automatically. Inside we found a device that gave us the ability to travel to any place on Earth at any time. Our first destination was Las Vegas. The Luxor hotel in Las Vegas is a huge square pyramid. The base measures 105 feet by 105 feet. You also know that the heights of the triangular sides from the top of the pyramid to the base (and bisecting the base) are 195 feet. As Joe and I walked into the hotel we saw a large group of people huddled in a circle. We then saw a poster that told us that this was a rope-climbing contest and four rope-climbing champions were competing in the event. The rope extended from the top of the pyramid straight down to the ground of the lobby. The overconfident climbers refused to climb unless the rope was over 175 feet, because they wanted a challenge after their past climb.

We told them how to solve this problem using the knowledge of the dimensions of the pyramid that were posted by the entrance. To do this, we first had to find the slant length of the pyramid, which was not posted on the information sign. We again used the Pythagoras Theorem. Side 1 was the height of the triangular side (195 feet.) Side 2 is 52.5 feet because the height bisects the base (105 / 2.) We could now determine the hypotenuse, which was the slant length. 195 squared + 52.5 squared = slant length squared. They found that the slant length of Luxor is 201.94 feet. As the rope-climbers stood puzzled we then used the Pythagoras theorem again. This time, the rope length was side 1, and we called it x. Side two was 52.5 because it was the length from the bottom of the rope extending straight to the base of the pyramid (bisecting the base.) The hypotenuse was the height of one of the triangular sides. We used the formula x squared + 52.5 squared = 195 squared. We found that the rope was 187.80 feet and the climbers could climb it.

Our next stop in Las Vegas was the Stratosphere. The Stratosphere is the tallest free-standing structure in North America. On the top there are some rides and attractions. As we got to the top, we saw some construction workers talking about a new feature of the stratosphere. It is going to be called the Spinning Pointer. It will a thin, steel triangle placed on the very top column on the stratosphere. Each vertex of the triangle will spray out a cool mist for tourists who are trying to beat the Las Vegas heat while on the Stratosphere. The problem is, the dim engineers want to find the most balanced point of the triangle to bolt it down with. They were going to merely guess and check to see if the massive steel triangle would find a balance point, but we told them exactly how to find it.

Obviously, these engineers had not passed Geometry in High School, because all they had to know about was the four points of concurrency on the Euler line. For this problem we taught them what the centroid of a triangle was. We told them to measure each side of the steel triangle. Then find the midpoint of each side. Finally, draw a line from each vertex to the midpoint of each vertex�s opposite side. We told them that the three medians of a triangle meet at a point, and that point is your center of gravity. The engineers took our advice and bolted it down to the top of the Stratosphere.

The engineers were not done. They then had to attach gold �Viva Las Vegas� tiles to the top of the steel triangle. The triangle came in from the factory with some of the tiles already attached to give the idea of the layout they wanted. The note said, �continue this pattern.� We looked at the steel triangle, and in the first row, there were 3 tiles, in the second row, there were 6 tiles, in the third row there were 11 tiles, and in the fourth row, there were 18 tiles. The engineers wanted to start from the bottom row of the triangle so they could do the more difficult work (with more tiles) first. However, they had no clue how many tiles should go in the last row. The triangle had 22 rows.

The first thing we did to help them was we recognized that the pattern was quadratic. We set up the following T-Chart: