Geometry & Trigonometry

A line modelling the stock returns, $y$, is expected to have an approximate linear relationship with spending ability, $x$. This can be modelled by the line $y = 3x + 1$.

A ship sails from point $P$ on a bearing of $310^\circ$ for $k \text{ km}$ to point $Q$. It then changes course and sails on a bearing of $225^\circ$ for $10 \text{ km}$ to point $R$.

A structural engineer is designing a triangular framework where the side $AC$ has a fixed length of $8 m$. The dimensions $x$ and $y$ are dependent on the height h of the framework.

Consider a triangle that has an area of $15 m^2$, with two sides measuring $5 m$ and $8 m$. It is given that the angle between these sides is acute.

A ship sails from point $A$ to point $B$ on a bearing of $045^\circ$ and then from point $B$ to point $C$ on a bearing of $120^\circ$. The distance from $A$ to $B$ is $100\text{ km}$ and from $B$ to $C$ is $150\text{ km}$.

A cyclist is tracking their progress along a designated route represented by the equation of a line $y = 4x + 2$. This equation illustrates the cyclist's position over time on a graph.

In the diagram below,

Building regulations in a city state that the maximum angle of elevation of a roof is $35$ degrees. A building has a footprint of $7 \text{cm}$ by $5 \text{cm}$. The roof must be an isosceles triangle-based prism with a vertical line of symmetry.

Consider the cuboid shown in the image below. The base of the cuboid is a square with sides of length $3 \text{ cm}$ and a height of $5 \text{ cm}$.

A company is designing a ramp to load cargo onto trucks. The ramp follows the slope of a line given by the equation $y = \frac{1}{2}x + 3$, where $y$ is the height in meters above ground level, and $x$ is the horizontal distance from the base of the ramp.