Answer:
A person saves 30.7 yards
Step-by-step explanation:
the path a person walks diagonally creates a right triangle with the lenght and the width of the rectangular piece of land. The diagonal is 90 yards which becomes the hypotenuse.
let <em><u>w</u></em> be the width, then <em><u>l</u></em> becomes the length, which is twice the width. Therefore, leg <em><u>l=2</u></em><u>a</u> and leg <u><em>w=</em></u><u>a</u>.
By the Pythagorean Theorem:
![c^{2}=l^{2}+w^{2}](https://tex.z-dn.net/?f=c%5E%7B2%7D%3Dl%5E%7B2%7D%2Bw%5E%7B2%7D)
Replacing the values:
![90^{2}=(2a)^{2}+a^{2}\\90^{2}=4a^{2}+a^{2}\\90^{2}=5a^{2}\\\\a=\sqrt{\frac{90^{2}}{5} } } =\frac{90}{\sqrt{5} }](https://tex.z-dn.net/?f=90%5E%7B2%7D%3D%282a%29%5E%7B2%7D%2Ba%5E%7B2%7D%5C%5C90%5E%7B2%7D%3D4a%5E%7B2%7D%2Ba%5E%7B2%7D%5C%5C90%5E%7B2%7D%3D5a%5E%7B2%7D%5C%5C%5C%5Ca%3D%5Csqrt%7B%5Cfrac%7B90%5E%7B2%7D%7D%7B5%7D%20%7D%20%7D%20%3D%5Cfrac%7B90%7D%7B%5Csqrt%7B5%7D%20%7D)
But that <u>a</u> is the value for the width, if a person walks the width and the length, it becomes:
the length plus the width=2a+a=3a.
So:
![a=\frac{90}{\sqrt{5} } \\\\3a=\frac{3(90)}{\sqrt{5} }](https://tex.z-dn.net/?f=a%3D%5Cfrac%7B90%7D%7B%5Csqrt%7B5%7D%20%7D%20%5C%5C%5C%5C3a%3D%5Cfrac%7B3%2890%29%7D%7B%5Csqrt%7B5%7D%20%7D)
Now, the difference between walking across the land and surrounding it, is:
![d=\frac{3(90)}{\sqrt{5} }-90=30.7476\\\\d=30.7](https://tex.z-dn.net/?f=d%3D%5Cfrac%7B3%2890%29%7D%7B%5Csqrt%7B5%7D%20%7D-90%3D30.7476%5C%5C%5C%5Cd%3D30.7)
We can conclude, a person saves 30.7 yards if walking across the land.