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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).
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).
Solution :
Given:
Dimensions of the cuboidal pen stand: Length ($l$) = $15$ cm, Breadth ($b$) = $10$ cm, Height ($h_{cuboid}$) = $3.5$ cm.
Number of conical depressions ($n$) = $4$.
Radius of each conical depression ($r$) = $0.5$ cm.
Depth (height) of each conical depression ($h_{cone}$) = $1.4$ cm.
To Find:
The volume of wood remaining in the entire pen stand.
Step 1: Calculate the volume of the cuboidal block.
The formula for the volume of a cuboid is $V_{cuboid} = l \times b \times h$.
$V_{cuboid} = 15 \text{ cm} \times 10 \text{ cm} \times 3.5 \text{ cm}$
$V_{cuboid} = 150 \times 3.5 = 525 \text{ cm}^3$
Step 2: Calculate the volume of one conical depression.
The formula for the volume of a cone is $V_{cone} = \frac{1}{3}\pi r^2 h$.
Using $\pi \approx \frac{22}{7}$:
$V_{cone} = \frac{1}{3} \times \frac{22}{7} \times (0.5)^2 \times 1.4$
$V_{cone} = \frac{1}{3} \times \frac{22}{7} \times 0.25 \times 1.4$
$V_{cone} = \frac{1}{3} \times 22 \times 0.25 \times 0.2$ [Since $1.4 / 7 = 0.2$]
$V_{cone} = \frac{1}{3} \times 22 \times 0.05 = \frac{1.1}{3} \text{ cm}^3$
Step 3: Calculate the total volume of four conical depressions.
$V_{total\_cones} = 4 \times V_{cone}$
$V_{total\_cones} = 4 \times \frac{1.1}{3} = \frac{4.4}{3} \text{ cm}^3$
$V_{total\_cones} \approx 1.4667 \text{ cm}^3$
Step 4: Calculate the volume of wood in the stand.
The volume of wood is the volume of the cuboid minus the volume of the four conical depressions.
$V_{wood} = V_{cuboid} - V_{total\_cones}$
$V_{wood} = 525 - \frac{4.4}{3}$
$V_{wood} = \frac{1575 - 4.4}{3}$
$V_{wood} = \frac{1570.6}{3}$
$V_{wood} = 523.5333... \text{ cm}^3$
Final Answer: The volume of wood in the entire stand is approximately $523.53 \text{ cm}^3$.
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).
- 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 $\pi = 3.14$.
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