. A gulab jamun, contains sugar syrup up to about 30% of its volume. Find approximately how much syrup would be found in 45 gulab jamuns, each shaped like a cylinder with two hemispherical ends with length 5 cm and diameter 2.8 cm (see Fig. 13.15)

Let's visualize the shape of Gulab jamun as shown below.

From the above figure, we can see that,

length of cylindrical part = Length of a Gulab jamun - 2 × radius of the hemispherical part

Also,

Diameter of Gulab jamun = diameter of the cylindrical part

Radius of cylindrical part = radius of hemispherical part

In order to find the volume of sugar syrup in 45 Gulab jamuns, we need to find the volume of all the gulab jamun.

As it’s given a Gulab jamun contains sugar syrup up to about 30% of its volume. We make an assumption to every Gulab jamun contains sugar syrup at 30% of its volume to simplify the calculation.

Volume of sugar syrup = 30% of the volume of 45 gulab jamuns.

Diameter of the Gulab jamun, d = 2.8 cm

Radius of cylindrical part = radius of hemispherical part r = d/2 = 2.8/2 cm = 1.4 cm

Length of cylindrical part, h = 5 cm - 2 × 1.4 cm = 2.2 cm

Volume of one Gulab jamun =volume of cylindrical part + 2 × volume of the hemispherical parts

= πr²h + 2 × 2/3 πr³

= πr²h + 4/3 πr³

= πr² (h + 4r/3)

= 22/7 × 1.4 cm × 1.4 cm × (2.2 cm + (4/3) × 1.4 cm)

= [22/7 × 1.4 cm × 1.4 cm × (12.2/3 cm)]

= 75.152/3 cm³

Volume of 45 Gulab jamuns = 45 × volume of one Gulab jamun

= 45 × 75.152/3 cm³

= 15 × 75.152 cm³

= 1127.28 cm³

Volume of sugar syrup in 45 Gulab jamuns = 30% of volume of 45 Gulab jamun

= 30/100 × 1127.28 cm³

= 338.184 cm³

Thus, the volume of sugar syrup in 45 cylindrical shaped gulab jamuns is 338 cm³ (approximately).