Q3:

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).

Given:

• Total number of gulab jamuns ($n$) = $45$.
• Shape of one gulab jamun: A cylinder with two hemispherical ends.
• Total length of the gulab jamun ($L$) = $5$ cm.
• Diameter of the gulab jamun ($d$) = $2.8$ cm.
• Sugar syrup content = $30\%$ of the total volume.

To Find:

The total volume of sugar syrup in $45$ gulab jamuns.

Step 1: Determine the dimensions of the cylindrical and hemispherical parts.

The radius ($r$) of the cylinder and the hemispheres is half of the diameter:

$r = \frac{d}{2} = \frac{2.8}{2} = 1.4$ cm.

The length of the cylindrical part ($h$) is the total length minus the radii of the two hemispherical ends:

$h = L - (r + r) = 5 - (1.4 + 1.4) = 5 - 2.8 = 2.2$ cm.

Step 2: Calculate the volume of one gulab jamun.

The volume of one gulab jamun ($V_{total}$) is the sum of the volume of the cylinder and the volumes of the two hemispheres:

$V_{total} = V_{cylinder} + 2 \times V_{hemisphere}$

$V_{total} = \pi r^2 h + 2 \times (\frac{2}{3} \pi r^3)$

$V_{total} = \pi r^2 (h + \frac{4}{3} r)$

Substituting the values ($r = 1.4$, $h = 2.2$, $\pi \approx \frac{22}{7}$):

$V_{total} = \frac{22}{7} \times (1.4)^2 \times (2.2 + \frac{4}{3} \times 1.4)$

$V_{total} = \frac{22}{7} \times 1.96 \times (2.2 + 1.8667)$

$V_{total} = 22 \times 0.28 \times (4.0667) = 6.16 \times 4.0667 \approx 25.05$ cm$^3$.

Step 3: Calculate the total volume of 45 gulab jamuns.

$V_{45} = 45 \times V_{total} = 45 \times 25.05 = 1127.25$ cm$^3$.

Step 4: Calculate the volume of sugar syrup.

The syrup is $30\%$ of the total volume:

$V_{syrup} = 30\% \times V_{45} = 0.30 \times 1127.25$

$V_{syrup} = 338.175$ cm$^3$.

Rounding to the nearest whole number as per standard approximation in such problems:

Final Answer: The total volume of sugar syrup in 45 gulab jamuns is approximately 338 cm$^3$.