Homework SourseA swimming pool is 50m long and 20m wide its depth decreases
A swimming pool is 50m long and 20m wide. its depth decreases linearly along the length from 9m to 1m. it is initally empty and is filled at a rate of 1 m^3/min. How fast is the water level rising 250 min after the filling begins ? How long will it take to fill the pool ?
Solution
The sloping part of the pool is filled first, followed by the \"rectangular\" part. As the sloping part of the pool rises 2m along its 50m length then for a height (h) of water the length is 50h/2 = 25h. The Sloping Volume = Width x Triangular cross section of water = 20 x ½bh = 20 x ½ x 25h x h = 250h² cu.ms The \"rectangular\" part Volume = 20 x 50 x 1 = 1000 cu.ms.......and this only starts to fill when h > 2. As the water flows in at 1m³/min then Water Volume = t, where t is the time in minutes. Then t = 250h² : h² = t/250 : h = t^½/v250 Then the rate of change of water level = dh/dt = ½t^-½/v250 = 1/(2v250t) When t = 250 then dh/dt = 1/2v(250 x 250) = 1/500m/s = 2 mm/s When h = 2 (the sloping part is filled) then t = 250 x 2² = 1000 mins and as the volume of the \"rectangular\" part = 1000 cu.ms then this also takes 1000 mins to fill : Giving a total of 2000 mins to fill the pool.
