Unit 35 PRACTICAL APPLICATIONS WITH RIGHT TRIANGLES

2 PROBLEMS STATED IN WORD FORM Procedure for solving right triangle problems stated in word form: 1. Sketch a right triangle based on the given information 2. Label the known parts of the triangle with the given values. Label the angle or side to be found 3. Follow the procedure for determining an unknown angle or side of a right triangle

3 ANGLES OF DEPRESSION AND ELEVATION Two terms commonly used in practical application problems from various occupational fields are the angle of depression and the angle of elevation. The illustration below shows what is meant by these two terms Angle of elevation Horizontal Line Angle of depression Line of sight

4 EXAMPLE A surveyor is to determine the height of a TV relay tower. The transit is positioned at a horizontal distance 20 meters from the foot of the tower. An angle of elevation of 46° is read in sighting the top of the tower. The height from the ground to the transit telescope is 1.70 meters. Determine the height of the tower to the nearest hundredth of a meter. Sketch a picture and label the parts

5 EXAMPLE (Cont) – Sketch a picture and label the parts – Solve: tan 46° = (T/20) so T = m – Now add the height of the transit: m m = meters An Line of sight 46° Tower (T) 20 m Transit

6 EXAMPLE Some problems require forming two or more right triangles by projecting auxiliary lines. Compute linear values to two decimal places. Compute the length of piece AB of the roof truss. AB 2.20m 10.30m 35.3° 28.5°

7 EXAMPLE (Cont) Compute the length of piece AB of the roof truss. Drop two vertical lines to create right triangles. AB 2.20m 10.30m 35.3° 28.5°

8 EXAMPLE (Cont) Compute the length of piece AB of the roof truss. AB 2.20m 10.30m 35.3° 28.5°

9 A right circular conical tank has a height of 4 m and a radius of 1.2 m. The tank is filled to a height of 3.7 m with liquid. How many liters of liquid are in the tank? Think in a 2 dimensional sense  See the right triangle created by the top side, and altitude  See the one created by the top of the water, side, and altitude

10 A right circular conical tank has a height of 4 m and a radius of 1.2 m. The tank is filled to a height of 3.7 m with liquid. How many liters of liquid are in the tank? We can work with trigonometry or similar triangles to solve this one Since we are working with trig we will go that direction although they are similar. In both triangles, the angle at the bottom of the vessel is the same.

11 A right circular conical tank has a height of 4 m and a radius of 1.2 m. The tank is filled to a height of 3.7 m with liquid. How many liters of liquid are in the tank? So by figuring out the red right triangle’s bottom angle we have the purple one as well

12 A right circular conical tank has a height of 4 m and a radius of 1.2 m. The tank is filled to a height of 3.7 m with liquid. How many liters of liquid are in the tank? Now that we know the angle is we can redo our last slides work That tells us that the water has a radius of 1.11m the.3m below the edge it is

13 A right circular conical tank has a height of 4 m and a radius of 1.2 m. The tank is filled to a height of 3.7 m with liquid. How many liters of liquid are in the tank? So we know the water is 3.7 m deep and has a radius of 1.11m at the top. The vessel is in a shape of a right cone That does not finish the question though

14 PRACTICE PROBLEMS 1. Measuring out a horizontal distance of 65 feet on the ground from the base of a tower and then sighting the top of the tower from that position results in an angle of elevation of 54°. Determine the height of the tower to the nearest hundredth of a foot. 2. A highway entrance ramp rises 38.4 feet in a horizontal distance of feet. Determine the angle of inclination of the ramp.

15 PRACTICE PROBLEMS (Cont) 3. A conveyor belt is used by Patty’s Packaging Company to lift packages. The most efficient operating angle is 33.25°. If the packages are elevated 4.75 feet, how long is the conveyor belt? 4. A flower bed shaped like a right triangle has sides of 5.4 feet, 3.2 feet, and 4.35 feet. What are the measures of the two acute angles formed by the bed in DMS?

16 PRACTICE PROBLEMS (Cont) 5. Determine the gauge dimension y of the V-block below. EF and GF are equivalent. Round to 3 decimal places.

17 PROBLEM ANSWER KEY feet ° feet 4. 36° 18’ and 53° 42’ in