Write an equation for the width of the pool in terms of x. & If the area of the path is 140m^2, find the dimensions of the pool.

Question

a rectangular swimming pool has a perimeter of 46 meters. The length of the pool is x meters. 1. Write an expression for the width of the pool in terms of x. 

 

A path of width (x-8) meters is to be built around the pool.

 

2. If the area of the path is 140 m^2, find the dimensions of the pool.

 

Answer 1

Perimeter = 2L + 2w

 

46 = 2L + 2w

46 = 2(L+w)

 

 23  = L+w

 

 The dimensions of the pool are x and 23-x <--- answer to part 1

 

The path adds 2(x-8)= 2x-16  meters to each dimension.

 

The dimensions of the pool and walkway are  2x-16+x = 3x-16 and 2x-16 + 23-x = x + 7

 

The entire area is then (3x-16)(x+7)

 

The area of the pool is x(23-x)

 

Entire Area = Area of pool + area of walkway

 

  (3x-16)(x+7) = x(23-x) + 140

 

  3x^2 + 21x - 16x - 112 = 23x - x^2 + 140

 

 3x^2 + 5x - 112 = 23x - x^2 + 140

 

 4x^2 -18x - 252 = 0

 

  2x^2 - 9x - 126 = 0

 

 x = [ 9 +or- sqrt( 81 + 4(2)(126))]/4

 

     = [ 9 +or- sqrt( 81 + 1008)]/4

 

      = [ 9 +or- sqrt(1089)]/4

 

      = [ 9 +or- 33]/4

 

the negative case results in negative measures

 

x = (9+33)/4 = 42/4 = 21/2 = 10.5

 

The dimensions of the pool 10.5 and 23-10.5 = 12.5

The width of the path is 10.5-8 = 2.5

 

CHECK:

The dimensions of the entire area is (10.5 + 2*2.5) = 10.5 +5 = 15.5 and 12.5 + 2*2.5 = 12.5 + 5 = 17.5

The entire area is 15.5 * 17.5 = 271.25

 

The area of the pool is 10.5 * 12.5 = 131.25

 

271.25-131.25 = 140

 

The results above in bold are correct!