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 $\pi = 3.14$.

Given:

• Shape of the vessel: A sphere attached to a cylindrical neck.
• Cylindrical neck dimensions: Height ($h$) = $8$ cm, Diameter ($d_1$) = $2$ cm.
• Spherical part dimensions: Diameter ($d_2$) = $8.5$ cm.
• Measured volume by the child = $345$ cm$^3$.
• Constant: $\pi = 3.14$.

To Find:

Whether the child's measurement of $345$ cm$^3$ is correct by calculating the actual volume of the vessel.

Step 1: Calculate the volume of the cylindrical neck.

The radius of the cylinder ($r_1$) is half of its diameter ($d_1$).

$r_1 = \frac{d_1}{2} = \frac{2 \text{ cm}}{2} = 1 \text{ cm}$.

The formula for the volume of a cylinder is $V_{cylinder} = \pi r_1^2 h$.

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

$V_{cylinder} = 3.14 \times 1 \times 8 = 25.12 \text{ cm}^3$.

Step 2: Calculate the volume of the spherical part.

The radius of the sphere ($r_2$) is half of its diameter ($d_2$).

$r_2 = \frac{d_2}{2} = \frac{8.5 \text{ cm}}{2} = 4.25 \text{ cm}$.

The formula for the volume of a sphere is $V_{sphere} = \frac{4}{3} \pi r_2^3$.

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

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

$V_{sphere} = \frac{12.56 \times 76.765625}{3}$

$V_{sphere} = \frac{964.21625}{3} = 321.3920833... \text{ cm}^3 \approx 321.39 \text{ cm}^3$.

Step 3: Calculate the total volume of the vessel.

Total Volume ($V_{total}$) = $V_{cylinder} + V_{sphere}$.

$V_{total} = 25.12 + 321.3920833...$

$V_{total} = 346.5120833... \text{ cm}^3$.

Step 4: Comparison and Conclusion.

The calculated volume is approximately $346.51$ cm$^3$.

The child measured the volume as $345$ cm$^3$.

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

Final Answer: The child is incorrect; the actual volume of the vessel is approximately 346.51 cm$^3$.