A rectangular field 250 m long by 110 m wide. How much shorter is it to walk diagonally across the field than around the sides?
It's 1/4 the time shorter.
Use Pythagoras
h^2 = 250^2 + 110^2
= 74600
h = √74600
= 273.13 m
so what is 250+110 - 273.13 ?
To answer this question, we need to calculate the distances of walking diagonally across the field and walking around the sides separately.
The diagonal of a rectangle can be found using the Pythagorean theorem, which states that in any right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In our case, the length, width, and diagonal form a right-angled triangle.
Let's calculate the diagonal first:
Using the Pythagorean theorem:
Diagonal^2 = Length^2 + Width^2
Substituting the given values:
Diagonal^2 = 250^2 + 110^2
Diagonal^2 = 62500 + 12100
Diagonal^2 = 74600
Diagonal = √74600 ≈ 273.19 m
Now let's calculate the distance of walking around the sides:
Perimeter = 2 × (Length + Width)
Substituting the given values:
Perimeter = 2 × (250 + 110)
Perimeter = 2 × 360
Perimeter = 720 m
Finally, to find how much shorter it is to walk diagonally across the field, we subtract the distance of the diagonal from the distance of walking around the sides:
Shorter distance = Perimeter - Diagonal
Shorter distance = 720 - 273.19
Shorter distance ≈ 446.81 m
Therefore, it is about 446.81 meters shorter to walk diagonally across the field than around the sides.