Question

A model of Leafy Ball Fountain is made to be kept on the tabletop. Water gently cascades down the ball into a decorative cylindrical pool where it is recycled.  The diameter of spherical ball is 21 cm. Cylindrical pool - Outer diameter is 50 cm and inner diameter is 40 cm. Height of solid base is 14 cm. Height of water filled is 7 cm.

Observe the figure and answer the following questions:

1. Determine the total height of the fountain.   [1]
2. Find the volume of the ball.   [1]
3. 1. If one-third of the ball is submerged in the water, find the volume of the water filled in the pool.   [2]
      OR
   2. Find the sum of the outer curved surface area of the cylindrical part and surface area of the ball.   [2]

Answer 1

Given: Diameter of spherical ball = 21 cm

Outer diameter of cylindrical pool = 50 cm

Inner diameter of cylindrical pool = 40 cm

Height of solid base = 14 cm

Height of water filled = 7 cm

(i) Radius of ball, r = `d/2`

= `21/2` 

= 10.5 cm

Total height = Height of solid base + Height of water + Radius of ball (since ball is on top)

= 14 + 7 + 10.5

= 31.5 cm

(ii) The volume of the ball.

 Volume of sphere = `4/3 pi r^3`

V = `4/3 pi (10.5)^3`

= `4/3 xx 22/7 xx 1157.625`

= `(4 xx 22 xx 1157.625)/21`

= `101871/21`

= 4851 cm³

(iii) (a) Inner radius of cylindrical pool = `40/2`

= 20

Volume of cylinder = πr²h

= `22/7 xx (20)^2 xx 7`

= `22/7 xx 400 xx 7`

= 22 × 400

= 8800 cm³

Volume of submerged part of ball = `1/3 xx "Volume of ball"`

= `1/3 xx 4851`

= 1617

Volume of water filled = 8800 − 1617

= 7183 cm³

(iii) (b) Outer radius cylinder = `50/2`

= 25

Curved surface area of cylinder = 2πrh

= `2 xx 22/7 xx 25 xx 14`

= 2200 cm²

Surface area of sphere = 4πr²

= `4 xx 22/7 xx (10.5)^2`

= `4 xx 22/7 xx 110.25`

= `(4 xx 22 xx 110.25)/7`

= `9702/7`

= 1386 cm²

Sum = 2200 + 1386

= 3586 cm²