Mathematical Topic: Coordinate Geometry

A Nuclear Disaster

A Nuclear Disaster

Chernobyl is the site of a nuclear disaster that happened in 1986.

Due to the slow decay of radioactive elements, the site is still extremely dangerous today.

In 2016, engineers covered the main damaged nuclear reactor with a giant concrete cap, as shown below:

A NUCLEAR DISASTER

The cross section of this cover can be modelled with the parametric equations:

 

\[x = 8\left( t + 10 \right)\ ,\ y = 100 – t^{2}\ ,\ – 19 \leq t \leq 10\]

 

a) Find the Cartesian equation of this model.

[3 marks]

b) The width of the cap, according to the model.

[2 marks]

An Advancing Hurricane

An Advancing Hurricane

A hurricane is heading from the Atlantic Ocean to Miami, Florida. If the centre of the hurricane goes within a 50 kilometre radius of the city centre, the city will sustain very severe damage.

A meteorologist models the city centre as a fixed particle located at the point (240 , 200). They then model the centre of the hurricane as travelling on a path relative to the city defined by \(y = kx + 20\). 

The unit of distance for the meteorologist’s coordinate system is kilometres.

 

a) Find the values of k such that the city does not sustain severe damage.

[8 marks]

b) State one limitation of the model.

[1 mark]

A Carbon Negative Company

A Carbon negative company

A small company’s carbon dioxide (CO₂) emissions since 2000 can be modelled using the parametric equations

\[12x = t,\ y = 8t – 4.9t^{2} + 10,\ t \geq 0\]

Where x is the number of years since 2000 and y the yearly CO₂ emissions, in tonnes. 

Due to an afforestation program, the company’s emissions can go negative, as the trees planted absorb CO₂ from the atmosphere. 

a) Find the year in which the company’s CO₂ emissions go negative.

[4 marks]

b) Find the greatest amount of CO₂ emissions from that company in one year

[5 marks]

Separated Satellites

Separated Satellites

2 weather-monitoring satellites orbit the earth.

One is in a circular orbit C₁, the other orbits in an extreme ellipse, C₂ so that it can get closer to the surface. 

These orbits can be modelled by the equations below:

\[C_{1}:\left( x + 2\sqrt{17} \right)^{2} + y^{2} = 66\]

\[C_{2}:x = 10\cos t,\ y = 4\sqrt{2}\sin t,\ 0 \leq t \leq 2\pi\]

Give the x coordinate of the points of intersection of the curves C₁ and C₂, given that

\(- 5 \leq x \leq 0\). Give an exact answer in the form

\[A\sqrt{1122} + B\sqrt{17}\]

[7 marks]

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