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.

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  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

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  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

option a, c,d are correct.

step-by-step explanation:

from the given figure, it is given that z is equidistant from the sides of the triangle rst, then from triangle tzb and triangle szb, we have

tz=sz(given)

bz=zb(common)

therefore, by rhs rule,δtzb ≅δszb

by cpctc, sz≅tz

also, from δctz and δasz,

tz=sz(given)

∠tcz=∠saz(90°)

by rhs rule, δctz ≅ δasz, therefore by cpctc, ∠ctz≅∠asz

also,from δasz and δzsb,

zs=sz(common)

∠zbs=∠saz=90°

by rhs rule, δasz ≅δzsb, therefore, by cpctc, ∠asz≅∠zsb

hence, option a, c,d are correct.