Right angle triangles (SOH CAH TOA): 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 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 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 rectangular prism with length 10 cm, width 8 cm, and height 6 cm is used as a base for a sculpture. One corner, vertex H, is cut off by a flat plane passing through points P, Q, and R. Point P is on edge HE such that HP = 3 cm. Point Q is on edge HG such that HQ = 4 cm. Point R is on edge HD such that HR = 3 cm.

(a) Calculate the volume of the remaining part of the prism.

(b) Calculate the total surface area of the remaining part of the prism.

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 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.

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 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.

(a) Two drones, P and Q, are launched simultaneously. Drone P is launched from the origin $(0,0)$ of a coordinate system, while Drone Q is launched from a point $15$ units along the positive horizontal axis. The velocity equations for the two drones are given by $\mathbf{v}_P$ and $\mathbf{v}_Q$ where:

$\mathbf{v}_P = \begin{pmatrix} 6 \\ 4 - 2t \end{pmatrix}$ and $\mathbf{v}_Q = \begin{pmatrix} -3 \\ 6 - 2t \end{pmatrix}$

Write down vector equations for the displacement, $\mathbf{r}_P$ and $\mathbf{r}_Q$, of the two drones at time $t$.

(b) Find an expression for the vector joining the centre of Drone P to the centre of Drone Q.

(c) Hence find the shortest distance between the two drones.

(d) The pilot of Drone P adjusts its initial velocity vector to $\begin{pmatrix} a \\ b \end{pmatrix}$. The velocity equation for Drone P is now $\mathbf{v}_P = \begin{pmatrix} a \\ b - 2t \end{pmatrix}$.

Find the values of $a$ and $b$ if the two drones collide when $t = 1.5$ seconds.

(e) Using the values of $a$ and $b$ found in part (d), find the angle to the horizontal at which Drone P was initially launched.

A large decorative water wheel, with a diameter of $40$ metres, has its lowest point $2$ metres above the water surface. There are eight evenly spaced buckets on the wheel. The wheel rotates in an anticlockwise direction.

Bucket A is initially located at the lowest point of the water wheel when it starts rotating. At the same time, Bucket B is located two positions ahead of Bucket A in the anticlockwise direction.

Let $t$ be the length of time, in minutes, after the water wheel started rotating.

(a) Calculate the height of Bucket B above the water surface when $t = 0$.

The water wheel completes one revolution every $10$ minutes.

(b) Calculate the height of Bucket A above the water surface when $t = 7.5$ minutes.

(c) Determine the values of $t$ when Bucket A and Bucket B have the same height above the water surface, for $0 \le t \le 10$.

The height of Bucket A above the water surface can be modelled by the function $h(t) = -R \cos(bt) + d$, for $t \ge 0$.

(d) State the amplitude of $h(t)$.

(e) Find the value of

(i) $b$, given that $b > 0$;

(ii) $d$.

A new attraction, 'Hydro-Spin', features a modified water wheel. A main beam, PQ, rotates in an anticlockwise direction about its midpoint O. The length of the beam is $30$ metres. The midpoint O is fixed at a height of $35$ metres above the water surface. At the end of the beam, point Q, there is a smaller circular wheel with a radius of $8$ metres, which also rotates anticlockwise.

A passenger, Clara, is in a bucket labelled K on the smaller wheel. The height of the end of the beam Q above the water surface, $T$ seconds after 'Hydro-Spin' starts moving, is modelled by the function $q(T) = 15 \sin(24T) + 35$. The height of bucket K above point Q, after $T$ seconds, is modelled by the function $k(T) = 8 \cos(36T)$.

(f) Find the furthest distance from Clara's bucket K to the midpoint O of the main beam.

(g) Calculate Clara's height above the water surface after $20$ seconds.

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 team of architects is designing a new domed stadium. The cross-section of the dome's roof can be modelled by the function $f(x) = 0.8e^{-0.5x^2}$, where $-3 < x < 3$. The $x$-axis represents the ground level and the $y$-axis represents the height above the ground, with units in metres.

A support beam is to be anchored to the dome's roof at a point, P, where the roof's height is $0.6$ m above the ground. The coordinates of P are $(a, 0.6)$, where $a > 0$.

Calculate the value of $a$.

Find an expression for $f'(x)$.

The support beam is designed to be perpendicular to the dome's roof at point P. Find the angle, $\theta$, that this support beam makes with the horizontal ground.

A circular park has a central monument. A straight walking path $AB$ cuts across the park. The radius of the park is $6$ m. The shortest distance from the center of the park $O$ to the path $AB$ is $5$ m.

(a) (i) Calculate the angle $A\hat{O}B$ in degrees.

(a) (ii) The region of the park bounded by the path $AB$ and the arc $AB$ is a flower bed. Calculate the area of this flower bed.

A square performance stage has a special effect light at its center. The stage has a side length of $10$ m. The light projects a circular pattern on the stage floor with a radius of $6$ m.

(b) (i) Calculate the total area of the circular light pattern if it were projected onto an infinite surface.

(b) (ii) Calculate the area of the stage floor that is lit by the special effect light.

Let $B$ be the brightness of the special effect light, measured in lumens, and $t$ be the time in minutes since the light was switched on.

The rate of change of brightness is given by $\frac{dB}{dt} = 0.5 t e^{-0.2t}$.

(c) Find the value of $t$ at which the brightness of the light is increasing at the greatest rate.