To determine the amount of syrup required for 45 gulab jamuns shaped like a cylinder with two hemispherical ends, we first need to calculate the volume of a single gulab jamun.
The formula for the volume of a cylinder is given by:
V_{cylinder} = πr^2h
And the volume of a hemisphere is:
V_{hemisphere} = (2/3)πr^3
Since each gulab jamun has two hemispherical ends, we can combine these formulas. The total volume (V) of one gulab jamun can be calculated as:
V = V_{cylinder} + 2 × V_{hemisphere}
Assuming the radius (r) of the gulab jamun is 2 cm and the height (h) of the cylindrical part is 3 cm:
– Volume of the cylinder:
V_{cylinder} = π(2^2)(3) = 12π cm³
– Volume of one hemisphere:
V_{hemisphere} = (2/3)π(2^3) = (16/3)π cm³
– Thus, the total volume for two hemispherical ends:
2 × V_{hemisphere} = (32/3)π cm³
Now, we can calculate the total volume of one gulab jamun:
V = 12π + (32/3)π = (36/3)π + (32/3)π = (68/3)π cm³
Now for 45 gulab jamuns:
Total Volume = 45 × (68/3)π = 1020π cm³
Using 3.14 for π, we can approximate this as:
Total Volume ≈ 1020 × 3.14 ≈ 3196.8 cm³
Therefore, approximately 3196.8 cm³ of syrup should be used for 45 gulab jamuns.