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.
It form's a right angle
Assuming x is the journey through west
And y is the journey through South
Leaving z the root back to sunshine from crismson
Where x=6 and y=8
Apply Pythagoras
x²+y²=z²
z=√(x²+y²)
If you want to reverse the journey and go back same way
Which one would be longer and which one would be shorter?
new road
the route is a right triangle with the hypotenuse as the direct road
... shorter than the sum of the E-W and N-S roads
6^2 + 8^2 = ?^2
D^2 = x^2 + y^2 = 6^2 + 8^2 = 100
D = 10 miles = Displacement or distance from starting point = shorter route.
d = x + y = 6 + 8 = 14 miles = Total distance traveled = longer route.
To determine whether taking the new road from Crimson back to Sunshine is the shortest route, let's analyze the distances involved.
From the given information, we know that you rode 6 miles due west from Sunshine to Happyville and then rode 8 miles south from Happyville to Crimson. Therefore, the total distance covered up until Crimson is 6 miles west + 8 miles south = 14 miles.
Now, let's consider the new road that goes directly from Crimson back to Sunshine. Since it is a straight road, we can use Pythagoras' theorem to calculate the distance of this route.
Using Pythagoras' theorem:
Distance^2 = (Distance west)^2 + (Distance south)^2
For the shorter route, the distance west is 6 miles, and the distance south is 8 miles.
Distance^2 = (6 miles)^2 + (8 miles)^2
Distance^2 = 36 + 64
Distance^2 = 100
Distance = 10 miles
So, the shorter route from Crimson back to Sunshine is 10 miles.
To determine whether this new road is shorter or not, we need to calculate the distance of the longer route, which involves going back the same way you came.
Since you traveled 14 miles from Sunshine to Crimson, the total distance for the longer route would be 14 miles * 2 (returning back along the same path) = 28 miles.
Therefore, the longer route would be 28 miles - 10 miles (shorter route) = 18 miles further than the shorter route.
In conclusion, if you want to take the shortest route back home, you should take the new road directly from Crimson to Sunshine, which is 10 miles long. Taking this route would be shorter by 18 miles compared to going back the way you came.