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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).
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).
Solution :
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$^3$.
More Questions from Class 10 Mathematics Surface Areas and Volumes EXERCISE 12.2
- Q1: A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to $1$ cm and the height of the cone is equal to its radius. Find the volume of the solid in terms of $\pi$.
- Q2: Rachel, an engineering student, was asked to make a model shaped like a cylinder with two cones attached at its two ends by using a thin aluminium sheet. The diameter of the model is $3$ cm and its length is $12$ cm. If each cone has a height of $2$ cm, find the volume of air contained in the model that Rachel made. (Assume the outer and inner dimensions of the model to be nearly the same.)
- Q4: A pen stand made of wood is in the shape of a cuboid with four conical depressions to hold pens. The dimensions of the cuboid are $15$ cm by $10$ cm by $3.5$ cm. The radius of each of the depressions is $0.5$ cm and the depth is $1.4$ cm. Find the volume of wood in the entire stand (see Fig. 12.16).
- Q5: A vessel is in the form of an inverted cone. Its height is $8$ cm and the radius of its top, which is open, is $5$ cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius $0.5$ cm are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.
- Q6: A solid iron pole consists of a cylinder of height $220$ cm and base diameter $24$ cm, which is surmounted by another cylinder of height $60$ cm and radius $8$ cm. Find the mass of the pole, given that $1$ cm$^3$ of iron has approximately $8$g mass. (Use $\pi = 3.14$)
- Q7: A solid consisting of a right circular cone of height $120$ cm and radius $60$ cm standing on a hemisphere of radius $60$ cm is placed upright in a right circular cylinder full of water such that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is $60$ cm and its height is $180$ cm.
- Q8: A spherical glass vessel has a cylindrical neck $8$ cm long, $2$ cm in diameter; the diameter of the spherical part is $8.5$ cm. By measuring the amount of water it holds, a child finds its volume to be $345$ cm$^3$. Check whether she is correct, taking the above as the inside measurements, and $\pi = 3.14$.
CBSE Solutions for Class 10 Mathematics Surface Areas and Volumes
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