Spaceship battle

Launch worksheet

Launch solution 

The Scenario

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You are the commanding general of the starship Ramilles, defending the earth from an invasion by evil robots from planet Proton. The robots have driven you all the way back to the solar system and you are defending earth against the onslaught.

Suddenly the alien robot mothership appears – this is your chance. If you can co-ordinate the laser beams from your two most powerful ships, you have a chance at taking down the mothership and saving all of humanity. You’ll need to wait until the mothership is in the perfect intersection point of the two beams.

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The linear equation for the beam from your first spaceship is:

\begin{aligned} y=-4x+27 \end{aligned}

The linear equation for the beam from second spaceship is:

\begin{aligned} y=2x-3 \end{aligned}

Approach

Here you have two linear equations, and you need to find the point at which they intersect. If the two lines intersect, they will have the same y value at the intersection point. So you can set the two equations to have the same y value and then calculate what the x value would be:

\begin{aligned} -4x + 27 = 2x - 3 \\ 27 + 3 = 2x + 4x \\ 30 = 6x \\ x = 5 \end{aligned}

This gives you the x coordinate where the beams will intersect, but not the y coordinate. You can substitute the x value back into either of the two equations to find out the y value:

\begin{aligned} y=2x-3 \\ y = 2 \times 5 - 3 \\ y = 7  \end{aligned}

There you have it – you need to wait until the alient mothership is at the (x, y) coordinates of (5, 7) before firing. You can also check your answer by graphing the two lines and seeing where they intersect on a graph:

excel_plot

From the graph it looks like the two lines intersect at (5, 7), just like we calculated. Let’s see what happens if we wait for the mothership to his those coordinates before firing:

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I’d say that’s mission accomplished!

Real Life Example – Mobile Phone Contracts

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A lot of things like internet and mobile phone plans can be bought on contracts of different lengths. Typically short term contracts have a big up front fee and then a recurring monthly fee. Longer term contracts often reduce or “waive” (remove) the initial fee because they offer the company selling them a longer term source of income from the customer.

For example, consider these two contracts:

Contract A: $200 upfront, and then $50 / month for a minimum of 12 months

Contract B: No fee upfront, and then $50 / month minimum for 24 months

Contract A earns the retailer a gross (total return, not just profit) guaranteed return of $200 + 12 x $50 = $800 from that customer

Contract B earns the retailer a gross guaranteed return of $0 + $50 x 24 = $1200

So although contract B has no upfront fee, the company is usually happy to offer it because they have a customer locked in for a longer period of time resulting in a larger guaranteed return to the company.