Surface Areas and Volumes | Exercise 12.2

Question 8

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 $345\text{ cm}^3$ . Check whether she is correct, taking the above as the inside measurements, and $\pi = 3.14$ .

Solution

Understand the Question

The glass vessel is composed of two geometric shapes: a sphere (the bottom body) and a cylinder (the top neck).
Total volume of the vessel is given by: $\text{Total Volume} = V_{\text{sphere}} + V_{\text{cylinder}}$
We calculate the volume using $\pi = 3.14$ with the respective radius and height, then compare the result with the child's measurement of $345\text{ cm}^3$ .

Step 1 · Find Dimensions and Volume of Spherical Part

Diagram 1

For the spherical part: $\text{Radius } (r_1) = \dfrac{8.5}{2} = 4.25\text{ cm}$

\begin{aligned} V_{\text{sphere}} &= \dfrac{4}{3} \pi r_1^3 \\[0.6em] &= \dfrac{4}{3} \times 3.14 \times (4.25)^3 \\[0.6em] &= \dfrac{4}{3} \times 3.14 \times 76.765625 \\[0.6em] &\approx 321.39\text{ cm}^3 \end{aligned}

Step 2 · Find Dimensions and Volume of Cylindrical Neck

For the cylindrical neck: $\text{Radius } (r_2) = \dfrac{2}{2} = 1\text{ cm}, \quad \text{Height } (h) = 8\text{ cm}$

\begin{aligned} V_{\text{cylinder}} &= \pi r_2^2 h \\[0.6em] &= 3.14 \times (1)^2 \times 8 \\[0.6em] &= 25.12\text{ cm}^3 \end{aligned}

Step 3 · Calculate Total Volume and Compare

\begin{aligned} V_{\text{total}} &= V_{\text{sphere}} + V_{\text{cylinder}} \\[0.6em] &= 321.39 + 25.12 \\[0.6em] &\approx 346.51\text{ cm}^3 \end{aligned}

Since the actual volume is approximately $346.51\text{ cm}^3$ , the child's measurement of $345\text{ cm}^3$ is incorrect.

Answer

No, she is not correct. The actual volume of the vessel is approximately $346.51\text{ cm}^3$ .

Common Mistakes

Diameter vs. Radius: Substituting diameter values ( $8.5\text{ cm}$ or $2\text{ cm}$ ) directly into the volume formula instead of halving them to get the radius.
Incorrect $\pi$ Value: Using $\pi = \dfrac{22}{7}$ instead of the given value $\pi = 3.14$ .
Cubing Fractions / Decimals: Arithmetic errors when calculating $(4.25)^3 = 76.765625$ .

More questions in Exercise 12.2

Unless stated otherwise, take $\pi = \dfrac{22}{7}$ .

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 $\pi$ .

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 $3\text{ cm}$ and its length is $12\text{ cm}$ . If each cone has a height of $2\text{ cm}$ , 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.)

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 $5\text{ cm}$ and diameter $2.8\text{ cm}$ (see Fig. 12.15).

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 vessel is in the form of an inverted cone. Its height is $8\text{ cm}$ and the radius of its top, which is open, is $5\text{ cm}$ . It is filled with water up to the brim. When lead shots, each of which is a sphere of radius $0.5\text{ cm}$ are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.

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 $1\text{ cm}^3$ of iron has approximately $8\text{ g}$ mass. (Use $\pi = 3.14$ )

A solid consisting of a right circular cone of height $120\text{ cm}$ and radius $60\text{ cm}$ standing on a hemisphere of radius $60\text{ cm}$ 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 $60\text{ cm}$ and its height is $180\text{ cm}$ .