Question

A spherical glass vessel has a cylindrical neck 7 cm long and diameter 2 cm, while the diameter of the spherical part is 8.4 cm. Find how much water can be filled in the vessel.

Hint:

When combining solid figures, simply add or subtract their volumes depending on the shape configuration. Always use consistent units.

Answer 1

Step 1: Given data.
Diameter of cylindrical neck \( = 2 \text{ cm} \Rightarrow r_1 = 1 \text{ cm} \)
Length of cylindrical neck \( = h = 7 \text{ cm} \)
Diameter of spherical part \( = 8.4 \text{ cm} \Rightarrow r_2 = 4.2 \text{ cm} \)

Step 2: Volume of water that can be filled.
The vessel consists of a cylinder + sphere. \[ \text{Total Volume} = \text{Volume of cylinder} + \text{Volume of sphere} \]
Step 3: Apply formulas.
\[ \text{Volume of cylinder} = \pi r_1^2 h = \pi (1)^2 (7) = 7\pi \text{ cm}^3 \] \[ \text{Volume of sphere} = \frac{4}{3} \pi r_2^3 = \frac{4}{3} \pi (4.2)^3 \] \[ (4.2)^3 = 74.088 \] \[ \text{Volume of sphere} = \frac{4}{3} \pi \times 74.088 = 98.784\pi \text{ cm}^3 \]
Step 4: Total volume.
\[ \text{Total Volume} = 7\pi + 98.784\pi = 105.784\pi \] \[ \text{Total Volume} = 105.784 \times 3.14 = 331.16 \text{ cm}^3 \] Step 5: Conclusion.
Hence, the total amount of water that can be filled in the vessel is \( \boxed{331.16\ \text{cm}^3} \).