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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 $\pi = 3.14$)
Solution :
Given:
A solid iron pole composed of two cylinders:
- Cylinder 1 (Base): Height ($h_1$) = $220$ cm, Diameter ($d_1$) = $24$ cm.
- Cylinder 2 (Top): Height ($h_2$) = $60$ cm, Radius ($r_2$) = $8$ cm.
- Density of iron: $1$ cm$^3$ = $8$ g.
- Constant: $\pi = 3.14$.
To Find:
The total mass of the iron pole in grams (or kilograms).
Step 1: Determine the dimensions of the cylinders.
For the base cylinder (Cylinder 1):
Radius ($r_1$) = $\frac{\text{Diameter}}{2} = \frac{24 \text{ cm}}{2} = 12$ cm.
Height ($h_1$) = $220$ cm.
For the top cylinder (Cylinder 2):
Radius ($r_2$) = $8$ cm.
Height ($h_2$) = $60$ cm.
Step 2: Calculate the volume of the pole.
The volume of a cylinder is given by the formula: $V = \pi r^2 h$.
Total Volume ($V_{total}$) = Volume of Cylinder 1 + Volume of Cylinder 2
$V_{total} = (\pi \cdot r_1^2 \cdot h_1) + (\pi \cdot r_2^2 \cdot h_2)$
$V_{total} = \pi [ (12)^2 \cdot 220 + (8)^2 \cdot 60 ]$
$V_{total} = 3.14 [ (144 \cdot 220) + (64 \cdot 60) ]$
$V_{total} = 3.14 [ 31680 + 3840 ]$
$V_{total} = 3.14 [ 35520 ]$
$V_{total} = 111532.8$ cm$^3$.
Step 3: Calculate the mass of the pole.
Given that $1$ cm$^3$ of iron has a mass of $8$ g.
Total Mass = $V_{total} \times 8$ g/cm$^3$
Total Mass = $111532.8 \times 8$
Total Mass = $892262.4$ g.
Step 4: Convert to kilograms (optional but standard).
Since $1000$ g = $1$ kg:
Total Mass = $\frac{892262.4}{1000} = 892.2624$ kg.
Final Answer: The mass of the pole is 892262.4 g or approximately 892.26 kg.
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.
- 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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