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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 $\pi = 3.14$.
Solution :
Given:
- Shape of the vessel: A sphere attached to a cylindrical neck.
- Cylindrical neck dimensions: Height ($h$) = $8$ cm, Diameter ($d_1$) = $2$ cm.
- Spherical part dimensions: Diameter ($d_2$) = $8.5$ cm.
- Measured volume by the child = $345$ cm$^3$.
- Constant: $\pi = 3.14$.
To Find:
Whether the child's measurement of $345$ cm$^3$ is correct by calculating the actual volume of the vessel.
Step 1: Calculate the volume of the cylindrical neck.
The radius of the cylinder ($r_1$) is half of its diameter ($d_1$).
$r_1 = \frac{d_1}{2} = \frac{2 \text{ cm}}{2} = 1 \text{ cm}$.
The formula for the volume of a cylinder is $V_{cylinder} = \pi r_1^2 h$.
$V_{cylinder} = 3.14 \times (1)^2 \times 8$
$V_{cylinder} = 3.14 \times 1 \times 8 = 25.12 \text{ cm}^3$.
Step 2: Calculate the volume of the spherical part.
The radius of the sphere ($r_2$) is half of its diameter ($d_2$).
$r_2 = \frac{d_2}{2} = \frac{8.5 \text{ cm}}{2} = 4.25 \text{ cm}$.
The formula for the volume of a sphere is $V_{sphere} = \frac{4}{3} \pi r_2^3$.
$V_{sphere} = \frac{4}{3} \times 3.14 \times (4.25)^3$
$V_{sphere} = \frac{4}{3} \times 3.14 \times 76.765625$
$V_{sphere} = \frac{12.56 \times 76.765625}{3}$
$V_{sphere} = \frac{964.21625}{3} = 321.3920833... \text{ cm}^3 \approx 321.39 \text{ cm}^3$.
Step 3: Calculate the total volume of the vessel.
Total Volume ($V_{total}$) = $V_{cylinder} + V_{sphere}$.
$V_{total} = 25.12 + 321.3920833...$
$V_{total} = 346.5120833... \text{ cm}^3$.
Step 4: Comparison and Conclusion.
The calculated volume is approximately $346.51$ cm$^3$.
The child measured the volume as $345$ cm$^3$.
Since $346.51 \text{ cm}^3 \neq 345 \text{ cm}^3$, the child's measurement is incorrect.
Final Answer: The child is incorrect; the actual volume of the vessel is approximately 346.51 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.)
- 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).
- 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.
CBSE Solutions for Class 10 Mathematics Surface Areas and Volumes
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