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?

Accepted Solution

Answer:A person saves 30.7 yardsStep-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:[tex]c^{2}=l^{2}+w^{2}[/tex]Replacing the values:[tex]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} }[/tex]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:[tex]a=\frac{90}{\sqrt{5} } \\\\3a=\frac{3(90)}{\sqrt{5} }[/tex]Now, the difference between walking across the land and surrounding it, is:[tex]d=\frac{3(90)}{\sqrt{5} }-90=30.7476\\\\d=30.7[/tex]We can conclude, a person saves 30.7 yards if walking across the land.