Question

You and your friend enjoy riding your bicycles. Today is a beautiful sunny day, so the two of you are taking a long ride out in the country side. Leaving your home in Sunshine, you ride 6 miles due west to the town of Happyville, where you turn south and ride 8 miles to the town of Crimson. When the sun begins to go down, you decide that it is time to start for home. There is a road that goes directly from Crimson back to Sunshine. If you want to take the shortest route home, do you take this new road, or do you go back the way you came? Justify your decision. How much further would the longer route be than the shorter route? Assume all roads are straight.

Answer 1

Ok so to answer this you need to use pythagorean theorem,
\(c^2=a^2+b^2\) (assumed right triangle by going west then south)

Now the road that goes straight from Crimson to Sunshine is your hypotenous and the roads to and from Happyville are your legs,
\(c^2=6^2+8^2\)

We can answer this by solving for it,
\(c^2=6^2+8^2\)
\(c^2=6(6)+8(8)\)
\(c^2=36+64\)
\(c^2=100\)
\(\sqrt{c^2}=\sqrt{100}\)
\(c=10\)

So the road from Crimson to Sunshine is 10 miles.

It asks to find the shortest route, either to Happyville then Sunshine (by the legs) or directly to Sunshine (hypotenous),
Legs = \(6+8\) = \(14\)
Hypo = \(10\)

So it would be faster to travel straight to Sunshine than it would be to travel to Happyville then Sunshine because going straight to Sunshine is 10 miles versus going to Happyville then Sunshine, which is 14 miles.

The longer route ended up being the 2 roads, which was 14 miles, and the shorter one was 10 miles, so you subtract the two,
\(14-10\) = \(4\)
The difference is 4 miles.