Question

A rectangular piece of land whose length is twice its width has a diagonal distance of 90 yards. How many​ yards, to the nearest tenth of a​ yard, does a person save by walking diagonally across the land instead of walking its length and its​ width?

Answer 1

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 w be the width, then l becomes the length, which is twice the width. Therefore, leg l=2a and leg w=a.

By the Pythagorean Theorem:

$c^{2}=l^{2}+w^{2}$

Replacing the values:

$90^{2}=(2a)^{2}+a^{2}\\90^{2}=4a^{2}+a^{2}\\90^{2}=5a^{2}\\\\a=\sqrt{\frac{90^{2}}{5} } } =\frac{90}{\sqrt{5} }$

But that a 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:

$a=\frac{90}{\sqrt{5} } \\\\3a=\frac{3(90)}{\sqrt{5} }$

Now, the difference between walking across the land and surrounding it, is:

$d=\frac{3(90)}{\sqrt{5} }-90=30.7476\\\\d=30.7$

We can conclude, a person saves 30.7 yards if walking across the land.