Applications of right & non-right triangles (bearings, pythagoras, elevation & depression): notes and practice questions

The diagram below shows a hot air balloon hovering at point H, $520$m vertically above a landing pad.

Point A is the point on the ground, directly below the hot air balloon.

An observer starts walking at a constant speed from point C towards point A. From point C, the observer looks upward at the hot air balloon at an angle of elevation of $30\degree$. After $20$ minutes, the observer is at point B and observes the same hot air balloon at an angle of elevation of $50\degree$.

Write down the size of the angle of depression from H to C.

Find the horizontal distance from A to C.

Calculate the distance the observer walked from C to B.

Determine the observer's average speed, in metres per hour.

A drone is used for aerial surveying. It starts at a central hub (H).

(a)(i) From H, it flies 15 km on a bearing of 040° to survey point S1. From S1, it flies 10 km on a bearing of 170° to survey point S2.

Determine the interior angle HS1S2.

(a)(ii) Determine the distance from S2 back to H.

(a)(iii) Determine the bearing the drone must use to travel directly from S2 to H.

(b) The drone's mission also includes a final delivery point D, such that S1S2DH forms a parallelogram. Write down the distance between point D and point H. Justify your answer.

A drone takes off from a launch pad located at the origin O(0, 0, 0) of a coordinate system. It flies in a straight line to a target destination T with coordinates (5.0, 3.0, 7.0). All units are in metres.

(a) Calculate the total distance the drone travels from the launch pad to the target.

(b) Determine the angle of elevation of the drone's flight path from the horizontal ground.

A modern architectural marvel, 'The Apex Tower', is designed with a square base and a single peak directly above the centre of the base. The side length of the square base is $240 \text{ m}$, and the vertical height of the tower from the centre of the base to its peak is $160 \text{ m}$.

Let the base be represented by square ABCD and the peak by V. Let M be the midpoint of the side AB.

(a) Calculate all the side lengths and angles of the triangle $\triangle VOM$, where O is the centre of the square base.

(b) Calculate all the side lengths and angles of the triangle $\triangle VAB$.

A monument features a prominent pyramidal cap. The length of the slant edge from the apex, A, to any corner of its square base is measured as 2.8 m, accurate to the nearest tenth of a meter. The side length of the square base is exactly 3.2 m. Let C be a corner of the base.

Write down the upper bound and lower bound for the possible lengths of edge AC.

Let H be the midpoint of one of the base edges. Determine the upper bound and lower bound for AH, the slant height of the pyramid's triangular faces.

For structural stability, the angle between the slant height (AH) and the base of the pyramid must be less than 35°. Show whether this monument's pyramidal cap meets this stability requirement. Justify your answer.

A drone A takes off from a control tower at 10:00. It flies north-east at a horizontal speed of $70\sqrt{2} \text{ kmh}^{-1}$ and climbs at a rate of $3 \text{ kmh}^{-1}$. At 10:00, it is at a height of $5 \text{ km}$ directly above the control tower.

Find an expression for the displacement of drone A from the control tower at time $t$ hours after 10:00. Assume the control tower is at the origin (0,0,0) and the positive y-axis points North, and the positive x-axis points East.

At 10:30, a second drone B is $12 \text{ km}$ directly above the control tower. It flies on a bearing of $300^\circ$ at a horizontal speed of $80 \text{ kmh}^{-1}$ and descends at a rate of $2 \text{ kmh}^{-1}$.

Find an expression for the displacement of drone B from the control tower $t$ hours after 10:00.

Find the distance the two drones are apart when they have the same height.

At 10:00 am, a reconnaissance drone is located 2 km East and 6 km North of a central control tower. A coordinate system is established with the control tower at the origin. The drone maintains a constant velocity of $\begin{pmatrix} -0.8 \\ -1.2 \end{pmatrix}$ kilometres per hour (km h$^{-1}$), where the components represent velocity in the East-West and North-South directions, respectively.

Write down an expression for the position vector $\mathbf{r}$ of the drone, $t$ hours after 10:00 am.

Find the time at which the bearing of the drone from the control tower is 180°.

In this question, $\mathbf{i}$ denotes a unit vector due east, and $\mathbf{j}$ denotes a unit vector due north.

Two drones, P and Q, are each flying with constant velocities.

The position vector of drone P, at time $t$ minutes, is given as $\mathbf{r}_P = (2 + 3t)\mathbf{i} + (5 - 2t)\mathbf{j}$.

The position vector of drone Q, at time $t$ minutes, is given as $\mathbf{r}_Q = (-1 + t)\mathbf{i} + (1 + 4t)\mathbf{j}$.

(a) Find the bearing on which drone P is flying.

(b) Find the value of $t$ when drone Q is directly west of drone P.

(c) Find the value of $t$ when drone Q is directly north-west of drone P.

A botanical garden is constructing a large planter box for exotic plants. The planter is shaped like an inverted frustum of a right pyramid, with a horizontal square top opening of side length $120$ metres and a smaller horizontal square base of side length $80$ metres.

The depth of the planter is $9$ metres.

Find the angle of inclination of the side walls of the planter to the horizontal.

(b) The point V is the theoretical vertex of the full pyramid from which the frustum is cut, and C is the centre of the square top opening.

(i) Find the total height of the pyramid from its theoretical vertex V to the centre of the top opening C.

(ii) Hence or otherwise, show that the volume of the planter is $91200 \text{ m}^3$.

The botanical garden orders $180000$ bags of a special soil mix, with each bag containing $0.5 \text{ m}^3$ of soil. Determine whether the ordered soil is sufficient to fill the planter.

To prevent soil erosion and water leakage, the entire interior surface of the planter (including the bottom base and the four side walls) needs to be lined with a waterproof membrane. Calculate the total area that needs to be lined.

(a) A new high-speed rail line is being constructed across a flat plain. A remote research facility, Aether Labs (A), is located 8 km from the nearest point on the rail line. A major distribution hub, Nexus Depot (N), is situated 17 km from Aether Labs, and the rail line passes directly through Nexus Depot. Aether Labs plans to build a straight access track to connect to the rail line at a junction J. Transport on the access track will operate at 80 km/h, and on the high-speed rail line at 200 km/h.

Determine the total time taken for transport from Aether Labs to Nexus Depot via junction J, in terms of the distance from J to Nexus Depot.

(b) Determine the total time taken for transport from Aether Labs to Nexus Depot via junction J, in terms of the angle between the access track AJ and the perpendicular line segment from Aether Labs to the rail line.

A geological survey team is setting up sensors in a remote area. A map of the area is represented on the following coordinate axes.

Three sensor locations are positioned at $S_1(0, 10)$, $S_2(16, 14)$ and $S_3(16, 0)$.

All measurements are in kilometres.

(a) Write down the distance between $S_2$ and $S_3$.

(b) Calculate the distance between $S_1$ and $S_2$.

(c) A geological team member is at sensor $S_2$ and needs to walk directly to sensor $S_1$. Calculate the bearing of $S_1$ from $S_2$.

A communication relay station is to be installed at a point that is an equal distance from each of the sensors at $S_1$, $S_2$, and $S_3$.

(i) Write down the gradient of the line segment $[S_1S_3]$.

(ii) Write down the coordinates of the midpoint of the line segment $[S_1S_3]$.

(iii) Hence, calculate the coordinates of the communication relay station.

Two lighthouses, Lighthouse A and Lighthouse B, are located along a coastline. Lighthouse A has a height of 60 m and Lighthouse B has a height of 45 m. A ship is positioned at sea such that its horizontal distance from the base of Lighthouse B is 1200 m.

Calculate the angle of elevation from the ship to the top of Lighthouse B.

The horizontal distance between the bases of Lighthouse A and Lighthouse B is 800 m.

Calculate the angle of depression from the top of Lighthouse A to the top of Lighthouse B.

A drone delivery service operates from three main hubs, P, Q, and R, whose locations are represented on a coordinate plane. All measurements are in kilometres.

Hub P is at $(0, 10)$, Hub Q is at $(8, 14)$, and Hub R is at $(8, 2)$.

Write down the distance between Hub Q and Hub R.

Calculate the distance between Hub P and Hub Q.

A drone is at Hub Q and needs to fly directly to Hub P. Calculate the bearing of P from Q.

A central charging station is to be built at a location equidistant from Hubs P, Q, and R.

Write down the gradient of the line segment [PR].

Find the coordinates of the midpoint of [PR].

Hence calculate the coordinates of the central charging station.

A high-altitude drone takes off from a research station, point P. The drone flies due east at a constant speed of 50 km/h. It departs at 08:00. At 10:00, it reaches point Q and changes its course. It then flies for another 1.5 hours at the same speed, reaching point R. The course adjustment at Q is 40° southward from its original eastward direction.

(a) Determine the distance of the drone from the research station (PR) at 11:30. Write down your answer correct to the nearest integer.

(b) Determine the angle R$\hat{P}$Q. Give your answer correct to one decimal place.

(a) A ground crew on Luna Prime is tracking a supply drone. From the Aether station's control tower, the drone is observed to be $8500$ km East and $4000$ km North in a localized flat-map approximation.

(i) Find the straight-line distance from the Aether station to the drone.

(ii) Find the bearing of the drone from the Aether station. Give your answer in degrees, correct to one decimal place.

(b) The Aether station (A) is located at the origin $(0,0,0)$ of a 3D Cartesian coordinate system for this part. A research probe (P) is located at $(4000, 0, 0)$ km. A navigation beacon (B) is located at $(0, 4000, 0)$ km.

(i) Show that the position vector of the research probe, $\vec{p}$, is perpendicular to the position vector of the navigation beacon, $\vec{b}$.

(ii) Assuming the probe and beacon are on the surface of Luna Prime with a radius of $4000$ km, calculate the shortest distance along the surface between the research probe and the navigation beacon.

(c) Consider the Aether station (A) at $(4000, 0, 0)$ km, the Boreas station (B) at $(0, 4000, 0)$ km, and a North Pole reference point (N) at $(0, 0, 4000)$ km. Let $\vec{a}$, $\vec{b}$, and $\vec{n}$ be their respective position vectors from the centre of Luna Prime.

(i) Find the vector $\vec{a} \times \vec{b}$.

(ii) Show that the angle at vertex A in the spherical triangle formed by Aether, Boreas, and the North Pole (ABN) is $90^{\circ}$.

(d) A supply route between Aether and a new outpost, Delta, has an arc length of $2000$ km. Given that the radius of Luna Prime is $4000$ km, show that the central angle $\theta$ between Aether and Delta is $28.6^{\circ}$, correct to three significant figures.

(e) The Aether station (A) is located at $40^{\circ}$ N, $20^{\circ}$ E, and the Boreas station (B) is located at $70^{\circ}$ N, $100^{\circ}$ E on Luna Prime, which has a radius of $6000$ km. Find the shortest distance along the surface from Aether to Boreas.

(f) Using the vector method from part (c), find the initial bearing from Aether to Boreas. Give your answer in degrees, correct to one decimal place.