Episode 4: Nautical Radar and quadratic equations

This week the topic was quadratic equations and their applications. We interviewed Colin Wright, who works on radar systems for coordinating and tracking ships and boats.

Interesting links:

• Quadratic equations in the real world
• Marine Radar on Wikipedia
• How to use the quadratic formula
• Path of a projectile (interactive GeoGebra page)

Puzzle: A boat is going to sail 20km upstream along a river, then 20km back to where it started. Due to the speed of water flowing in the river, its speed is reduced by 2kph on the way upstream and increased by 2kph on the way downstream.

If the speed of the boat’s engine is x (kph), this means it travels 20km at (x-2)kph and then 20km at (x+2)kph. The total journey needs to take 3.5 hours. What value of x does the boat driver need to use?

Solution:

Since speed = distance / time we can rearrange that to get time = distance / speed.
Then our speed is (x-2) on the way out and (x+2) on the way back, so
20/(x-2) + 20/(x+2) = 3.5
Multiply through by (x+2) and (x-2):
20(x+2) + 20(x-2) = 3.5(x+2)(x-2)

Now expand the brackets to get a quadratic equation: 3.5x^2 - 40x - 14 = 0.
We can use the quadratic formula to get two solutions (one positive and one negative), but we know that x is positive, as it’s a speed so x=11.77kph.