A circular swimming pool is surrounded by a concrete wall 4 ft. wide. If the area of the concrete wall surrounding the pool is $$\frac{{11}}{{25}}$$ that of the pool, then the radius of the pool is :
Correct Answer: 20 ft
Let the radius of the pool be R ft.
Radius of the pool including the wall = (R + 4)ft.
Area of the concrete wall :
$$\eqalign{ & = \pi \leftsq.ft \cr & = \pi \leftsq.ft \cr & = 8\pi \left( {R + 2} \right)sq.ft \cr & 8\pi \left( {R + 2} \right) = \frac{{11}}{{25}}\pi {R^2} \cr & \Rightarrow 11{R^2} = 200\left( {R + 2} \right) \cr & \Rightarrow 11{R^2} - 200R - 400 = 0 \cr & \Rightarrow 11{R^2} - 220R + 20R - 400 = 0 \cr & \Rightarrow 11R\left( {R - 20} \right) + 20\left( {R - 20} \right) = 0 \cr & \Rightarrow \left( {R - 20} \right)\left( {11R + 20} \right) = 0 \cr & \Rightarrow R = 20 \cr} $$
∴ Radius of the pool = 20 ft.
Radius of the pool including the wall = (R + 4)ft.
Area of the concrete wall :
$$\eqalign{ & = \pi \leftsq.ft \cr & = \pi \leftsq.ft \cr & = 8\pi \left( {R + 2} \right)sq.ft \cr & 8\pi \left( {R + 2} \right) = \frac{{11}}{{25}}\pi {R^2} \cr & \Rightarrow 11{R^2} = 200\left( {R + 2} \right) \cr & \Rightarrow 11{R^2} - 200R - 400 = 0 \cr & \Rightarrow 11{R^2} - 220R + 20R - 400 = 0 \cr & \Rightarrow 11R\left( {R - 20} \right) + 20\left( {R - 20} \right) = 0 \cr & \Rightarrow \left( {R - 20} \right)\left( {11R + 20} \right) = 0 \cr & \Rightarrow R = 20 \cr} $$
∴ Radius of the pool = 20 ft.
