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

The vessel is made of two parts — a sphere (the round body) and a cylinder (the neck). To check the child's answer, we find the actual volume by adding the volume of the sphere and the volume of the cylinder, then compare it with $345\text{ cm}^3$ .

Step 1 — Find the dimensions

The radius is always half of the diameter, so $r = \dfrac{\text{diameter}}{2}$ .

Spherical part — diameter $= 8.5\text{ cm}$ :

$r_1 = \frac{8.5}{2} = 4.25 \text{ cm}$

Cylindrical neck — diameter $= 2\text{ cm}$ :

$r_2 = \frac{2}{2} = 1 \text{ cm}$

The length of the neck is its height, so $h = 8\text{ cm}$ .

Diagram 1

Step 2 — Volume of the spherical part

The volume of a sphere is $\dfrac{4}{3}\pi r^3$ . Using $\pi = 3.14$ and $r_1 = 4.25\text{ cm}$ :

$V_{\text{sphere}} = \frac{4}{3} \pi r_1^3$

$V_{\text{sphere}} = \frac{4}{3} \times 3.14 \times (4.25)^3$

$V_{\text{sphere}} = \frac{4}{3} \times 3.14 \times 76.765625$

$\boxed{V_{\text{sphere}} \approx 321.39 \text{ cm}^3}$

Step 3 — Volume of the cylindrical part

The volume of a cylinder is $\pi r^2 h$ . Using $\pi = 3.14$ , $r_2 = 1\text{ cm}$ and $h

$V_{\text{cylinder}} = \pi r_2^2 h$

$V_{\text{cylinder}} = 3.14 \times (1)^2 \times 8$

$\boxed{V_{\text{cylinder}} = 25.12 \text{ cm}^3}$

Step 4 — Total volume of the vessel

Add the two parts:

$V_{\text{total}} = V_{\text{sphere}} + V_{\text{cylinder}}$

$V_{\text{total}} = 321.39 + 25.12$

$\boxed{V_{\text{total}} \approx 346.51 \text{ cm}^3}$

Step 5 — Compare and conclude

The actual volume is about $\mathbf{346.51\text{ cm}^3}$ , but the child measured $\mathre not equal, so the child's measurement is not correct.

Answer

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

Question 8

Answer

More questions in Exercise 12.2