Answer:
The equation is [tex]\\ \frac{5}{2} : \frac{10.5ft}{x}[/tex]; The value of x is 4.2ft.
Step-by-step explanation:
A ratio is like a constant that remains between two values, and we can use it to find whatever others that keep the same constant relation between them.
Hank wants a dog run that keeps a constant relation between length to the width. That is, the length must be 2.5 times to the width ( [tex]\\ \frac{5}{2} = 2.5[/tex] ).
So, knowing that ratio or constant, we can represent it as follows:
[tex]\\ \frac{lenght}{width} : \frac{5}{2} \\[/tex] [ 1 ]
But, it also could be expressed as the relation between the width to the length:
[tex]\\ \frac{width}{length}:\frac{2}{5}[/tex] [ 2 ]
He wants a lenght of 10.5ft for building a dog run for his dog, and that this new value must keep the ratio just explained [ 1 ] to the width expected.
So, the equation is:
[tex]\\ \frac{5}{2} : \frac{10.5ft}{x}[/tex]
And we have to find the value for x that solve this equation.
However, we can use an easier way to represent this using the equation [ 2 ] for solving x :
[tex]\\ \frac{w}{l} :\frac{2}{5} : \frac{x}{10.5ft} \\\\ x = \frac{2 * 10.5ft}{5}=4.2ft\\[/tex]
That is, the width must be 4.2ft to keep the ratio length to the width 5:2 ( or the ratio width to the length 2:5).
To check this answer:
[tex]\\ \frac{length}{width} : \frac{5}{2} =2.5[/tex]
[tex]\\ \frac{length}{width} = \frac{10.5ft}{4.2ft} = 2.5[/tex].
[tex]\\ \frac{width}{length} : \frac{2}{5} = 0.4\\[/tex]
[tex]\\ \frac{width}{length} = \frac{4.2ft}{10.5ft} = 0.4[/tex].
