Question

The length of a rectangular pool is to be four times its width and a sidewalk 6ft wide will surround the pool. If a total area of 1600 square feet has been set aside for the area of the pool and the sidewalk, what are the dimensions of the pool?

Answer 1

Answer:

52 feet by 13 feet.

Step-by-step explanation:

Given:

The length of a rectangular pool is to be four times its width.

Sidewalk is 6 feet wide will surround the pool.

If a total area of 1600 square feet has been set aside for the area of the pool and the sidewalk.

Question asked:

What are the dimensions of the pool?

Solution:

Let width of a rectangular pool = $x$

Then the length of a rectangular pool will be = $4x$

$Area\ of \ rectnagle= length\times breadth$

                              $=x\times4x\\ \\ =4x^{2}$

Combined length of pool and the sidewalk = length of pool + width of sidewalk + width of sidewalk

Combined length of pool and the sidewalk = $4x+6+6=4x+12$

Similarly, Combined width of pool and the sidewalk = $x+6+6=x+12$

Combined area of pool and the sidewalk = 1600

$Combined\ lengthh\times Combined\ r=breadth=1600\\ \\ (4x+12)(x+12)=1600\\ \\ 4x(x+12)+12(x+12)=1600\\ \\ 4x^{2} +48x+12x+144=1600\\ \\ 4x^{2} +60x+144=1600$

Subtracting both sides by 144

$4x^{2} +60x=1456\\ \\ 4x^{2} +60x-1456=0\\ \\ Taking\ 4\ as\ common\\ \\ x^{2} +15x-364=0\\ \\ x^{2} +28x-13x-364=0\\ \\ x(x+28)-13(x+28)=0\\ \\ x+28=0,x-13=0\\ \\ x=-28,x=13$

Since dimensions can never be in negative, hence $x=13$

Width of a rectangular pool = $x$ = 13 feet

The length of a rectangular pool will be = $4x$ = $4\times13=52\ feet$

Therefore, the dimensions of the pool are 52 feet by 13 feet.