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

A landscape architect is designing a new garden. A straight path is to be built, and its boundary can be modelled by the equation $3x + 2y - 24 = 0$, where $x$ and $y$ are distances in metres.

(a) Write down the equation of the line in the form $y = mx + c$.

The garden's main feature is a triangular flower bed, with vertices at the origin O(0,0), and the points where the path intersects the $x$-axis (point A) and the $y$-axis (point B).

(b) Given that the line intersects the $x$-axis at point A and the $y$-axis at point B, find the coordinates of A and B.

A landscape architect is designing a new garden. A straight path is to be built, and its boundary can be modelled by the equation $3x + 2y - 24 = 0$, where $x$ and $y$ are distances in metres. The garden's main feature is a triangular flower bed, with vertices at the origin O(0,0), and the points where the path intersects the $x$-axis (point A) and the $y$-axis (point B).

(c) Calculate the area of triangle OAB.

(a) Calculate the shortest distance from the apex $V$ to the side $AB$ of the pyramid's base.

(b) Find the total surface area of the four triangular glass panels forming the pyramid's roof.

(c) Calculate the total height of the monument from the ground to the apex $V$.

(d) Determine the size of the angle between the slant edge $VA$ and the base edge $AB$.

(e) A security camera is installed at a point $S$ on the vertical edge $C'C$ of the monument. A guard standing at point $B'$ (a corner of the base of the monument) measures the angle of elevation to point $S$ to be $40^\circ$. Find the height of the security camera from the ground.

(a) A surveyor is measuring a tall communications mast. From a point $P$ on the ground, the angle of elevation to the top of the mast is $38^\circ$. The total height of the mast from the ground to its top is $70$ meters. Calculate the horizontal distance $d$ from point $P$ to the near edge of the mast's base, giving your answer correct to one decimal place.

(b) Calculate the direct line-of-sight distance $L$ from point $P$ to the top of the mast, giving your answer correct to one decimal place.

(c) A maintenance platform is located $15$ m below the top of the mast. Find the angle of depression from this maintenance platform to point $P$, giving your answer in degrees correct to one decimal place.

A designer is creating a symmetrical display stand for a museum exhibit. The cross-section of the stand is an isosceles trapezium $ABCD$, where $BC$ is parallel to $AD$. The shorter parallel side $BC$ measures $15$ cm, the longer parallel side $AD$ measures $27$ cm, and the non-parallel sides $AB$ and $CD$ each measure $10$ cm.

Show that the height of the trapezium is $8$ cm.

Hence, find the area of the cross-section of the display stand.

Find the size of the angle $ADC$, giving your answer to one decimal place.

Calculate the length of the diagonal $AC$, giving your answer to three significant figures.

(a) A glamping tent is designed in the shape of a rectangular-based pyramid. The base of the tent measures $8$ m by $6$ m, and the vertical height of the tent from the centre of the base to the apex is $4$ m.

Calculate the volume of the tent.

(b) The tent is advertised to accommodate a certain number of people, with each person requiring $3.2 \text{ m}^3$ of air space. Using your answer from part (a), calculate the maximum number of people the tent can accommodate.

(c) Show that the length of the longest sloping edge of the tent is $6.40$ m, correct to 3 significant figures.

(d) Find the angle at the apex between two adjacent longest sloping edges, specifically the angle formed by the two sloping edges that meet at the apex and span the $6$ m width of the base. Give your answer in degrees, correct to one decimal place.

(e) Calculate the total surface area of the tent fabric, excluding the base. Give your answer correct to one decimal place.

A drone flies from a launch pad L on a bearing of $300^\circ$. It flies for $12 \text{ km}$ to a point M. From M, it changes direction and flies $12 \text{ km}$ to a communications tower T, which is located due west of the launch pad L.

(a) Find the distance from the launch pad L to the tower T.

(b) Find the bearing of the tower T from the point M.

The height, $h$ metres, of a passenger on a Ferris wheel $t$ minutes after the ride starts is given by $h(t) = a \cos(bt) + d$. The ride starts at $t=0$ when the passenger is at the lowest point. The lowest point is 1 metre above the ground. The diameter of the wheel is 20 metres. The wheel completes one revolution in 16 minutes.

(a) Find the maximum height reached by the passenger and hence find the value of $d$.

(b) Find the value of $a$.

(c) Show that $b = \frac{\pi}{8}$.

(d) The function $h(t)$ is transformed to a new function $g(t)$. The graph of $h$ is first translated by the vector $\begin{pmatrix} 4 \\ -1 \end{pmatrix}$ and then reflected in the $t$-axis. Find the equation of $g(t)$.

(a) The main antenna of a communication tower is located at point $A(2, 8, 10)$. The centre of the rectangular base of the tower is at point $C(0, 5, 0)$. A vertical support beam connects the antenna to the centre of the base. Calculate the length of this support beam, $AC$.

(b) The rectangular base of the tower has dimensions $6$ m by $4$ m. Calculate the length of the diagonal of this base.

(c) A support cable runs from the antenna $A$ to one of the corners of the base, say point $P$. Find the size of the angle that this support cable $AP$ makes with the base platform.