Tom has a backyard that is 100 ft by 60 ft.  He plans to install a rectangular swimming pool bordered by a concrete walkway of uniform width.  He wants the area of the pool to take up 1/2 of the...

Tom has a backyard that is 100 ft by 60 ft.  He plans to install a rectangular swimming pool bordered by a concrete walkway of uniform width.  He wants the area of the pool to take up 1/2 of the area of the entire backyard.  

Determine the width of the walkway (x). Round your answer to 2 decimals.  

Enter answer in ft. 

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lemjay | High School Teacher | (Level 3) Senior Educator

Posted on

First, solve for the total area of the backyard.

`A_t=100*60=6000`

Hence, the total area of the backyard is `6000ft^2` .

Then, take half of it to get the area of the pool.

`A_p=1/2A_t=1/2*6000=3000`

Hence, the area of the pool is `3000 ft^2` .

Also, if x is the width of the walkway, then the dimension of the pool are:

length`=100-2x`

width`=60-2x`

(See image for your reference.)

So, the area of the pool can be express as:

`A_p=(100-2x)(60-2x)`

To solve for the value of x, plug-in value of area of pool.

`300=(100-2x)(60-2x)`

Then expand right side.

`3000=6000-320x+4x^2`

Set left side equal to zero.

`0=4x^2-320x+3000`

To simplify the equation further, divide both sides by 4.

`0=x^2-80x+750`

Then, apply quadratic formula.

`x=(-b+-sqrt(b^2-4ac))/(2a)=-(-80+-sqrt((-80)^2-4(1)(750)))/(2*1)`

`x=40+-5sqrt34`

`x=40+5sqrt34=69.15`

`x=40-5sqrt34=10.85`

Consider only the value x=10.85, since the other value of x would result to a negative length and width of the swimming pool.

Therefore, the width of the walkway is 10.85 ft.

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