An Introductory example of Solving Three Dimensional Problem by using Trigonometry
Find the height of the cliff.
Example of a Three Dimension 3D Problem
The bases on a baseball diamond are 27.4 m apart. The pitcher pitches. and the batter hits a fly ball straight up 18 m. What is the maximum angle of elevation of the ball. to the nearest degree, as seen by the pitcher if he is standing at the centre of the diamond?
A pyramid has a square base of side length
10.5 m. The slant height of the pyramid is
a) Determine the height of the pyramid, to the nearest tenth of a metre.
b) Determine the angle
\theta between one face and the base, to the nearest tenth of a degree.
A square-based tent has the cross- sectional shape shown. The side wall goes up at an angle of elevation of 55° for 2.8 m. then continues at an angle of elevation of 32° for another 2.1 m to the peak.
a) Determine the height of the tent.
b) Determine the side length of the base.
c) Determine the length of one of the diagonals of the base.
A gift box for a perfume bottle is in the shape of a pyramid with a rectangular base. The dimensions of the base are 10 cm by 7 cm and the length of each side edge is 16 cm. Determine the height of the pyramid.
Beni and Alan watch a rocket as it is launched. Beni is 0.8 km closer to the launching pad than Alan.
When the rocket disappears from view, its angle of elevation for Beni is 36.5° and for Alan is 31.9°. Determine the altitude of the rocket, to the nearest tenth of a kilometre, at the time it disappears from View.
A rectangular prism has length 10 cm, width 8 cm. and height 5 cm.
a) Determine the length of the diagonal AB.
b) Determine the measure of angle
Angela is a surveyor. To find the height,
h, of an inaccessible cliff, she takes measurements at point A some distance from the cliff and at a second point, C. which is along the base of the cliff and 400 m from point A. Angela determines the angle of elevation from point C to the top of the cliff is 18°. She finds that the angle from point B, which is at the base of the cliff directly below the top, to C to A
is 35°, and that the angle from B to A to C is 27°.
a) Draw a diagram to represent this situation.
b) What is the height of the cliff, to the nearest metre?
Annette and Devon want to determine how high their model rocket can fly. The rocket is launched from point B and reaches an altitude,
h. Annette is standing at point A and measures the angle of elevation to the tip of the rocket at its highest point (D) to be 41°. Devon is standing at point C, 300 m from Annette, and measures the angle of elevation to be 30°. If
\angle BAC is 42.6° and
\angle BCA is 25.2°, what altitude does the rocket reach, to the nearest metre?
Kendra is flying in a hot-air balloon and notices a barn directly below the balloon and a farmhouse located at an angle of depression of 28°. After the balloon rises vertically a further 58 m, the angle of depression to the farmhouse is 42°.
a) How high is the balloon before it rises?
b) How far is the barn from the farmhouse?
To determine the height,
h, of a tree in Algonquin Park, a conservation officer records measurements in a diagram. Determine the height of the tree.
A hydro tower located on the side of a hill casts a 55-m-long shadow down the hill when the angle of elevation of the sun is 48.3°. Determine the height of the tower if the angle of inclination of the hill is 21.4°.