Applications of Right Triangles

1.) Finding the angle of a roof given the pitch/slope

The angle of a roof is extremely important for many reasons. The first main reason is the engineering and structural design. Specific angles are used to maximize the strength of the roof so it can hold additional weight other than just the roofing tiles. This is especially true in northern and Midwestern states where snowfall collecting on a roof can add an enormous amount of weight. A second reason is drainage of rainwater. If a roof does not have a proper angle, water will not drain correctly causing possible issues with water leaking into attics and second story rooms.

A roof’s pitch, or slope, is given by the roof’s vertical change divided by its horizontal change. For example, the roof below has what is called a 7/12 pitch. The vertical change is 7 feet and the horizontal change is 12 feet.

(Example of a 7/12 roof pitch)

Problem: Find the angle (rounded to the nearest hundredth) formed by the roof and a horizontal line by the following roof pitches:

a.) 4/12 pitch (standard roof pitch)

b.) 7/12 pitch (shown above)

c.) 2/12 pitch (considered low pitch)

Applications of Right Triangles

2.) Using angles of elevation to find the angle measure of ramps

Ramps are one of mankind’s greatest innovations. These marvels of physics and motion allow humans to transport themselves and large objects to higher elevations than would normally be possible. One example of this would be ramps up to buildings so they become accessible to those with physical disabilities. Over the course of history, ramps have been used to build some of our greatest achievement.

Historians believe ramps were used in one of the greatest human accomplishments, the Egyptian pyramids. Scholars believe ancient Egyptians used these ramps to move unbelievable large stones to heights unthinkable in those times. Given the sheer size of these blocks, some of which measured 4 ft by 4 ft by 8 ft and weighed over 9 tons (18,000 pounds!!), no other way has been discovered for moving them to such high levels.

Problem: Creating the ramps that were most likely used to build the pyramids would have been a construction marvel all by themselves. We can assume the angle of the ramps would have been low to

ease the burden of moving these 9 ton stones. A safe guess for the angle measure used is . Using this

information, how long would the ramps have needed to be to reach the 3 heights shown below on the Great Pyramid of Giza? Round your answers to the nearest foot.

Applications of Right Triangles

3.) Using vectors to find locations

Vectors are numbers that contain both a magnitude (size) and direction. One example of a vector is wind speed. When wind speeds are given on the news, they are given with both a speed and a direction. An example of this is a weatherman on TV saying, “The winds are gusting to 30 miles per hour out of the southwest.” The 30 mph represents the magnitude and the southwest represents the direction. Locations of boats in the ocean often use vectors and use directions such as north, south, east, and west. Given a fixed location, boats can be located if you know what speed and direction they have been taking.

Problem:A boat starts off at a seaport in New York City and travels at a angle southeast. After travelling 100 miles, the boat turns due east and travels another 100 miles. If the port wants to send a radio signal to the boat, at what angle will they need to send the signal and what distance will the signal need to travel to reach the boat? Round both values to the nearest tenth.

Applications of Right Triangles

4.) Finding the actual size of a television

Televisions are advertised in many stores and have become much larger over the past few years. But many people do not realize the false advertising that goes into selling the size of a television. When a TV is advertised as being 43 inches, they are not talking about the length of width of the TV. They are actually giving you the diagonal distance from a lower corner to an upper corner. This means the TVs are not as large as most people assume. Answer the following question knowing this information. Include a diagram and label the appropriate dimensions.

(Example of a 32 inch TV)

Problem:

Part 1: Several new big screen TVs are advertised as being 55 inches. If the length along the bottom of the TVs measures between 40 and 46 inches, what are the minimum and maximum possible dimensions for the height of the TVs? Round your answers to the nearest tenth of an inch.

Part 2: Newer televisions have an aspect ratio of 16:9. This means the ratio of the width to the height of the TV is 16:9. Find the dimensions of a 55 inch TV with a 16:9 aspect ratio. List your answer in inches rounded to the nearest tenth.