L2 Algebra
Question Bank
The main cable of a suspension bridge forms a parabola, with dimensions in metres as shown in the diagram below.
(i) By placing the vertex of the parabola at the origin $(0,0)$, find the equation that models the shape of the cable.
(ii) Find the height of the main cable above the roadway at a point 150 metres from the center of the bridge.
A logo is formed by two parabolas, similar to the one in the diagram. The vertices are at $(0, 4)$ and $(0, -4)$. The logo is 8 units wide at its widest point (the x-axis).
Find the equations of both the red (opening down) and blue (opening up) parabolas.
The cross-section of a satellite dish is a parabola with its vertex at the origin. Its dimensions in metres are shown in the diagram below. The equation of the parabola is of the form $y = ax^2$.
Find the width of the dish at a point where its depth is exactly half of the maximum depth.
The cross-section of a satellite dish is a parabola with its vertex at the origin. Its dimensions in metres are shown in the diagram below. The equation of the parabola is of the form $y = ax^2$.
Find the width of the dish at a point where its depth is exactly half of the maximum depth.
The entrance to a garden is a parabolic stone archway. The arch is 6 metres wide at the base and reaches a maximum height of 4.5 metres. A hook for a lantern is to be placed on the arch at a point that is 1 metre horizontally from the base on one side. Use algebra to find the height of the lantern hook above the ground.
The entrance to a tunnel is a parabolic arch that is 8 metres wide at the base and has a maximum height of 5 metres. A rectangular truck is 3 metres wide and 4 metres high. Will the truck be able to pass through the tunnel? Justify your answer with algebraic working.
The cross-section of a valley is modelled by the parabola $h = k(x-150)^2 + 10$, where $h$ is the height in metres above sea level and $x$ is the horizontal distance in metres from a survey point. The valley is 200 metres wide at a height of 90 metres. A horizontal bridge is to be built across the valley at a height of 50 metres above sea level. Find the length of the bridge.
A boat sits in a parabolic-shaped canal. The canal is 2 metres deep with a width of 2.56 metres at water level. The boat has a trapezoidal cross-section with a top width of 2.7 metres and a bottom width of 1.8 metres. The boat is 1 metre tall and sits 0.8 metres below the surface of the water, with its top 0.2 metres above the water surface. Will the boat fit in the canal? Justify your answer with algebraic working.
