Question

A circular pool of radius r m has a path 2 m wide around its perimeter. If the area of the pool is four-fifths of the total area, prove that r²-16r-16=0. Hence calculate the radius of the pool in metres.

Answer 1

9514 1404 393

Answer:

  pool radius is 8+4√5 ≈ 16.94 meters

Step-by-step explanation:

Let r represent the radius of the pool. Then r+2 is the radius to the outside of the walkway. The ratio of areas is given as 4/5, so we have ...

  πr²/(π(r+2)²) = 4/5

  5r² = 4(r² +4r +4) . . . . cross-multiply

  r² -16r -16 = 0 . . . . . . . subtract the right-side expression from both sides

__

  r² -16r +64 = 80 . . . . . add 80 to complete the square

  (r -8)² = 80

  r -8 = √80 = 4√5 . . . . square root; use only the positive root

  r = 8 +4√5 ≈ 16.94 . . . meters

The radius of the pool is 8+4√5 ≈ 16.94 meters.

_____

The area of a circle of radius r is given by the formula ...

  A = πr²