4.1 Right Triangle Applications

4.1 Right Triangle Applications Recommended Homework: 4.1: 13-21 odd, 23-28, 31, 32, 34, 38 2.3: 76, 83, 84

STRATEGIES:  Draw a picture  Label what you know  Label what you want to find with a variable  Find an equation representing what you want to know with what you want to find. (You may need to use more than one equation and substitute!!).  Solve the above equation  Make sure you have answered all questions asked and included units!

Terminology Angle of elevation- always measured from the horizon up

Angle of depression- always measured from the horizon down

NOTE: Neither of these can ever be measured from the vertical!

1 4.1 Right Triangle Applications SOLVING TRIANGLES: Strategy:  Find all sides o If 2 sides given use Pythagorean Theorem o If 1 side and an acute angle are given, use trig functions.

 Find all angles o Sum of the angle measures in a triangle = 180º o If you are given 2 sides and no acute angles, then use trig functions

Example 1: Solve the following triangle

2 4.1 Right Triangle Applications

3 4.1 Right Triangle Applications Example 2: Solve the triangle

Applications involving right triangles

Example 3: A safety regulation states that the maximum angle of elevation for a rescue ladder is 72º. If the fire departments longest ladder is 140 ft, what is the maximum safe rescue height?

4 4.1 Right Triangle Applications Example 4: At person is standing 20 ft from a wall where a rectangular painting is hanging. The angle of elevation is to the bottom of the painting is 4º. The angle of elevation to the top of the painting is 10º. How tall is the painting?

Example 5: Suppose you are standing 78 ft from the base of a 160ft tree and looking up at the top of the tree. Find the angle of elevation.

5 4.1 Right Triangle Applications Example 6: A passenger airplane flying at 30,000 ft observes 2 towns directly to the left of the airplane. The angles of depression to the 2 towns are 20º and 50º respectively. How far apart are the two towns?

6 4.1 Right Triangle Applications Example 7: What if one of the first town was on the left and the second was on the right?

7 4.1 Right Triangle Applications Example 8: Suppose the angles of depression to two towns directly to the left of a plane are 23º and 46º respectively. If the 2 towns are 48650 ft apart, what is the altitude of the plane?

8 4.1 Right Triangle Applications Example 9: At a point 250 ft from the base of a building, the angle of elevation to the bottom of a smoke stack is 38º while the angle of elevation to the top of the smoke stack is 52º. Find the height of the smoke stack.

9 4.1 Right Triangle Applications Example 10: A pilot in a helicopter looks out his window and sees a plane to his left 7 mi away at an angle of depression of 32º and another plane on his right with an angle of depression of 48º. How far apart are the two planes if they both are at the same elevation?

10 4.1 Right Triangle Applications Bearings In surveying and navigation, directions are often given in bearings. A bearing is the measure of the acute angle a path or line of site makes with the north-south line.

Example:

N 25º E N 60º W S 45º E

11 4.1 Right Triangle Applications Example 11: Two fire towers are 30 km apart, where tower A is due west of tower B. A fire is spotted from the towers and the bearings from A and B are N 76º E and N 56º W respectively. Find the distance of the fire from the line segment AB

12 4.1 Right Triangle Applications Example 12: A boat leaves the dock at 8am traveling with a bearing of N48ºE at 10 knots. At 10am, it turns 90º toward the north and travels at 15 knots. What is the bearing of the ship from the port at 1 pm?

What is the bearing of the port from the ship at 1pm?

What is the bearing of the ship from the spot where it makes the first turn?