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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.
Solution :
Given:
1. A vessel in the shape of an inverted cone with height $h_c = 8$ cm and radius $r_c = 5$ cm.
2. The vessel is filled with water to the brim.
3. Lead shots are spherical in shape with radius $r_s = 0.5$ cm.
4. When lead shots are dropped, $\frac{1}{4}$ of the water in the cone flows out.
To Find:
The number of lead shots ($n$) dropped into the vessel.
Step 1: Calculate the volume of the conical vessel.
The formula for the volume of a cone is $V_c = \frac{1}{3}\pi r_c^2 h_c$.
$V_c = \frac{1}{3} \times \pi \times (5)^2 \times 8$
$V_c = \frac{1}{3} \times \pi \times 25 \times 8$
$V_c = \frac{200}{3}\pi \text{ cm}^3$
Step 2: Determine the volume of water that flows out.
According to the problem, the volume of water that flows out is equal to $\frac{1}{4}$ of the total volume of the cone.
$V_{out} = \frac{1}{4} \times V_c$
$V_{out} = \frac{1}{4} \times \frac{200}{3}\pi = \frac{50}{3}\pi \text{ cm}^3$
Step 3: Calculate the volume of one spherical lead shot.
The formula for the volume of a sphere is $V_s = \frac{4}{3}\pi r_s^3$.
$V_s = \frac{4}{3} \times \pi \times (0.5)^3$
$V_s = \frac{4}{3} \times \pi \times 0.125$
$V_s = \frac{4}{3} \times \pi \times \frac{1}{8} = \frac{1}{6}\pi \text{ cm}^3$
Step 4: Formulate the equation to find the number of lead shots ($n$).
The volume of water displaced is equal to the total volume of the $n$ lead shots dropped into the vessel [By Archimedes' Principle].
$n \times V_s = V_{out}$
$n \times (\frac{1}{6}\pi) = \frac{50}{3}\pi$
Step 5: Solve for $n$.
Divide both sides by $\pi$:
$n \times \frac{1}{6} = \frac{50}{3}$
Multiply both sides by 6:
$n = \frac{50}{3} \times 6$
$n = 50 \times 2$
$n = 100$
Final Answer: The number of lead shots dropped in the vessel is 100.
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).
- 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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