1.) a ladder of length 4.5m leans against a wall making an angle 50degrees with the horizontal if the bottom of a window is 4m above the ground, what is the distance between the top of the ladder and the bottom of the window
2.) from a horizontal distance of 8.5km, a pilot observes that the angle of depression of the top and base of a control tower are 30degrees and 33degrees respectively. calculate correct to three significant figures. (a)the shortest distance between the pilot and the base of the control tower (b)the height of the control tower
1.) To find the distance between the top of the ladder and the bottom of the window, we can use trigonometry. The ladder, the wall, and the ground form a right-angled triangle.
Let x be the distance between the top of the ladder and the bottom of the window. Then we have:
cos(50 degrees) = x / 4.5m
x = 4.5m * cos(50 degrees)
x ≈ 4.5m * 0.6428
x ≈ 2.89m
Therefore, the distance between the top of the ladder and the bottom of the window is approximately 2.89m.
2.) Let the height of the control tower be h meters. The pilot, the top of the control tower, and the base of the control tower form a right-angled triangle.
(a) To find the shortest distance between the pilot and the base of the control tower, we can use the tangent function:
tan(33 degrees) = h / 8.5km
h = 8.5km * tan(33 degrees)
h ≈ 8.5km * 0.6494
h ≈ 5.51km
Therefore, the shortest distance between the pilot and the base of the control tower is approximately 5.51km.
(b) The total height of the tower will be the sum of h and the height from the base of the control tower to the top of the tower (which can be found using the tangent function and the given angle of depression):
tan(30 degrees) = h / x
x = h / tan(30 degrees)
x ≈ 5.51km / 0.5774
x ≈ 9.54km
The height of the control tower is approximately 9.54km.