A hemispherical tank full of water is emptied by a pipe at the rate of 3 4 7 litres per second. How much time will it take to empty half the tank , if it is 3 m in diameter?

Radius of base of tank = D 2 = 3 2 m Volume of tank = 2 3 πr 3 = 2 3 × 22 7 × 3 2 3 = 99 14 m 3 = 99 14 × 1000 l i t r e s = 99000 14 l i t r e s Volume of water to be emptied = 1 2 × volume of tank = 1 2 × 99000 14 l i t r e s = 99000 28 l i t r e s Time taken to empty 3 4 7 = 25 7 l i t r e s = 1 s e c Time taken to empty 1 l i t r e s = 7 25 s e c Time taken to empty 99000 28 l i t r e s = 99000 28 × 7 25 = 69300 700 = 990 s e c = 990 60 m i n = 16 . 5 m i n Time taken to empty half tank is 16 minute and 5 second.

A hemispherical tank full of water is emptied by a pipe at the rate of 3 4 7 litres per second. How much time will it take to empty half the tank , if it is 3 m in diameter?

Radius of base of tank = D 2 = 3 2 m Volume of tank = 2 3 πr 3 = 2 3 × 22 7 × 3 2 3 = 99 14 m 3 = 99 14 × 1000 l i t r e s = 99000 14 l i t r e s Volume of water to be emptied = 1 2 × volume of tank = 1 2 × 99000 14 l i t r e s = 99000 28 l i t r e s Time taken to empty 3 4 7 = 25 7 l i t r e s = 1 s e c Time taken to empty 1 l i t r e s = 7 25 s e c Time taken to empty 99000 28 l i t r e s = 99000 28 × 7 25 = 69300 700 = 990 s e c = 990 60 m i n = 16 . 5 m i n Time taken to empty half tank is 16 minute and 5 second.

A cone of base radius 4 cm is divided into two parts by drawing a plane through the mid-point of its height and parallel to its base. Compare the volume of the two parts.

Given, radius of base of cone = 4 cm The cone is divided in two parts through the height of the cone. Let, the height of the cone is h cm . So, the volume of the cone = 1 3 × π × 4 2 × h cm 3 = 16 πh 3 cm 3 ∴ Height of the smaller cone is h 2 cm and radius is 4 2 cm = 2 cm ∴ Volume of smaller cone = 1 3 π × 2 2 × h 2 cm 3 = 2 πh 3 cm 3 ∴ Volume of lower frustum of the cone = Total volume of cone - Volume of shorter cone = 16 πh 3 - 2 πh 3 cm 3 = 16 πh - 2 πh 3 cm 3 = 14 πh 3 cm 3 ∴ Ratio of two parts of the cone = 2 πh 3 : 14 πh 3 = 2 : 14 = 1 : 7 The ratio of volumes of two parts of the cone is 1 : 7 .

A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in figure. If the height of the cylinder is 10 c m , and its base is of radius 3.5 c m . Find the total surface area of the article.

From the given diagram it can be clearly observed that radius of hemisphere is same as radius of cylinder. Given radius of cylindrical part = radius of hemispherical part = r = 3 . 5 cm . Also, height of the cylinder h = 10 cm . The total surface area of the article = ( C S A of cylinder) + ( 2 × CSA of hemisphere) = ( 2 π r h ) + ( 2 × 2 π r 2 ) = ( 2 π × 3 .5 × 10 ) + ( 2 × 2 π × 3 .5 × 3 .5 ) = 70 π + 49 π = 119 × 22 7 = 17 × 22 = 374 cm 2