In the picture given below, one can see a rectangular in-ground swimming pool installed by a family in their backyard. There is a concrete sidewalk around the pool of width x m. The outside edges of the sidewalk measure 7 m and 12 m. The area of the pool is 36 sq. m. a. based on information given form a quadratic equations in terms of x.b. find the width of sidewalk around the pool​

The quadratic equation in terms of x is 4x²-38x+48 = 0 and the width of the sidewalk is 1.5 m.

Given,

The outer edges of the swimming pool are 7m and 12 m.

The width of the sidewalk is x m.

The area of the pool = 36 sq. m

To Find,

The quadratic equation in terms of x, and the width of the sidewalk.

Solution,

Since the width of the sidewalk is x m.

So the dimensions of the pool will be (7-2x) and (12-2x).

Now,

Area of the pool = 36 sq. m

(7-2x)(12-2x) = 36

84-14x-24x+4x² = 36

4x²-38x+48 = 0

4(x-8)(x-3/2) = 0

x = 8, 1.5

The value of x can't be 8 as it will make the dimensions of the pool negative

So, the value of x is 1.5 m.

Hence, the quadratic equation in terms of x is 4x²-38x+48 = 0 and the width of the sidewalk is 1.5 m.

Answer:

The quadratic equation in 'x' is 2x² -19x +24 = 0

Width of the sidewalk = 1.5m

Explanation:

Given,

Area of the pool = 36sq.m.

width of the sidewalk = 'x'm

Length of the pool including the sidewalk = 7m

Breadth of the pool including the sidewalk = 12m

Required to find,

Quadratic equation in terms of 'x'

Width of the sidewalk

Formula to be used

Area of the rectangle = length x breadth

Length of the pool = 7 -2x

Breadth of the pool = 12 -2x

Area of the pool = (7-2x)(12-2x)

(7-2x)(12-2x) = 36

84 - 14x -24x +4x² = 36

4x² - 38x + 48 = 0

2x² - 19x + 24 = 0

Hence,

The quadratic equation in 'x' is  2x² - 19x + 24 = 0

To find 'x'

To solve the quadratic equation by splitting the middle term, we need to find two numbers such that their sum = -19 and product = 48

Two such numbers are = -16 and -3

2x (x-8) - 3(x-8) = 0

(2x-3)(x-8) = 0

2x -3 = 0 or x = 8

x =[tex]\frac{3}{2}[/tex]  = 1.5 or x = 8

x = 8, is not possible because the length of the pool including the sidewalk is 7m.

So x = 1.5m

∴ Width of the sidewalk = 1.5m