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. 12.15).
Step-by-Step Solution
Key Concept: The jamun is a composite solid (cylinder + two hemispheres). Find its total volume using the formulae for volume of a cylinder $V_{cyl}=\pi r^{2}h$ and volume of a sphere $V_{sphere}=\frac{4}{3}\pi r^{3}$ (two hemispheres make a sphere). Then take 30% of this volume for syrup and multiply by 45.
1. Identify dimensions\
- Diameter $d = 2.8\,\text{cm}$ \=> radius $r = \frac{d}{2}=1.4\,\text{cm}$.\
- Total length of the jamun $L = 5\,\text{cm}$.\
- Length of the cylindrical part $h = L - d = 5-2.8 = 2.2\,\text{cm}$ (because the two hemispherical ends together contribute a length equal to the diameter).\
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2. Volume of the cylindrical part\
$$V_{cyl}=\pi r^{2}h = \pi (1.4)^{2}(2.2) = \pi \times 1.96 \times 2.2 = \pi \times 4.312 \approx 3.1416 \times 4.312 \approx 13.55\,\text{cm}^{3}.$$\
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3. Volume of the spherical part (two hemispheres form a sphere)\
$$V_{sphere}=\frac{4}{3}\pi r^{3}=\frac{4}{3}\pi (1.4)^{3}=\frac{4}{3}\pi \times 2.744 \approx \frac{10.976}{3}\pi \approx 3.6587\pi \approx 11.49\,\text{cm}^{3}.$$\
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4. Total volume of one gulab jamun\
$$V_{jamun}=V_{cyl}+V_{sphere}=13.55+11.49 \approx 25.04\,\text{cm}^{3}.$$\
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5. Volume of sugar syrup in one jamun (30% of total volume)\
$$V_{syrup\,per\,jamun}=0.30\times V_{jamun}=0.30\times 25.04 \approx 7.512\,\text{cm}^{3}.$$\
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6. Total syrup in 45 jamuns\
$$V_{syrup\,total}=45\times 7.512 \approx 338.04\,\text{cm}^{3}.$$\
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7. Result\
Since $1\,\text{cm}^{3}=1\,\text{mL}$, the amount of syrup is approximately $338\,\text{mL}$ (or $\approx 338\,\text{cm}^{3}$).
Correct Answer: Approximately $338\,\text{cm}^{3}$ (or $338\,\text{mL}$) of sugar syrup.