Question

A juice glass is cylindrical in shape with a hemispherical raised-up portion at the bottom. The inner diameter of the glass is 10 cm and its height is 14 cm. Find the capacity of the glass. (use π = 3.14)

Answer 1

Step 1: Understanding the problem:

We are given a juice glass with a cylindrical shape and a hemispherical raised-up portion at the bottom. The inner diameter of the glass is 10 cm, and its total height is 14 cm. We need to find the capacity of the glass, which is the volume of the liquid it can hold. The volume of the glass is the sum of the volume of the cylindrical portion and the volume of the hemispherical portion.

Step 2: Calculating the volume of the cylindrical portion:

The volume \( V_{\text{cylinder}} \) of a cylinder is given by the formula:
\[ V_{\text{cylinder}} = \pi r^2 h \] where:
- \( r \) is the radius of the base of the cylinder,
- \( h \) is the height of the cylindrical portion.
Given that the diameter of the glass is 10 cm, the radius \( r = \frac{10}{2} = 5 \) cm. The height of the cylindrical portion is the total height minus the height of the hemispherical portion. Since the height of the hemispherical portion is equal to its radius (5 cm), the height of the cylindrical portion is:
\[ h_{\text{cylinder}} = 14 - 5 = 9 \, \text{cm} \] Now, calculate the volume of the cylindrical portion:
\[ V_{\text{cylinder}} = 3.14 \times (5)^2 \times 9 = 3.14 \times 25 \times 9 = 3.14 \times 225 = 706.5 \, \text{cm}^3 \]

Step 3: Calculating the volume of the hemispherical portion:

The volume \( V_{\text{hemisphere}} \) of a hemisphere is given by the formula:
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi r^3 \] where:
- \( r \) is the radius of the hemisphere.
Since the radius of the hemisphere is the same as the radius of the base of the cylinder, \( r = 5 \) cm. Now, calculate the volume of the hemispherical portion:
\[ V_{\text{hemisphere}} = \frac{2}{3} \times 3.14 \times (5)^3 = \frac{2}{3} \times 3.14 \times 125 = \frac{2}{3} \times 392.5 = 261.67 \, \text{cm}^3 \]

Step 4: Calculating the total capacity of the glass:

The total capacity of the glass is the sum of the volumes of the cylindrical portion and the hemispherical portion:
\[ V_{\text{total}} = V_{\text{cylinder}} + V_{\text{hemisphere}} = 706.5 + 261.67 = 968.17 \, \text{cm}^3 \]

Conclusion:

The capacity of the glass is approximately \( 968.17 \, \text{cm}^3 \).