(a)
In the diagram, AOB is a straight line. < AOC = 3(x + y)°, < COB = 45°, < AOD = (5x + y)° and < DOB = y°. Find the values of x and y.
(b) From two points on opposite sides of a pole 33m high, the angles of elevation of the top of the pole are 53° and 67°. If the two points and the base are on te same horizontal level, calculate, correct to three significant figures, the distance between the two points.
Explanation
(a) (3(x + y) + 45 = 180°) ( sum of angles on a straight line)
(3x + 3y + 45 = 180 implies 3x + 3y = 135)
(3x = 135 – 3y implies x = 45 – y)
Also,
(5x + y + y = 180)
(5x + 2y = 180….. (*))
Put x = 45 – y into (*) above,
(5(45 – y) + 2y = 180 implies 225 – 5y + 2y = 180)
(225 – 3y = 180 implies 3y = 45)
(y = frac{45}{3} = 15°)
(x = 45 – y)
(x = 45 – 15 = 30°)
(b)
Let the distance BD = x and CD = y.
In (Delta ABD),
(tan 53 = frac{33}{x})
(x = frac{33}{tan 53})
= ( 24.868 cm)
In (Delta ACD),
(tan 67 = frac{33}{y})
(y = frac{33}{tan 67})
(y = 14.01 cm)
(therefore BC = x + y )
(24.868 + 14.01 = 38.878 cm)
(approxeq 38.9 cm)