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: Problem Definition:

A juice glass, shaped as a cylinder with a hemispherical base, has an internal diameter of 10 cm and a total height of 14 cm. The objective is to determine the glass's capacity, which is equivalent to its total volume. The total volume is the sum of the volumes of the cylindrical section and the hemispherical section.

Step 2: Cylindrical Volume Calculation:

The volume of a cylinder is calculated using \( V_{\text{cylinder}} = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height. Given the diameter 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. The hemisphere's height equals its radius (5 cm). Therefore, the height of the cylindrical portion is \( h_{\text{cylinder}} = 14 - 5 = 9 \) cm. The volume of the cylindrical portion is: \[ V_{\text{cylinder}} = 3.14 \times (5)^2 \times 9 = 3.14 \times 25 \times 9 = 3.14 \times 225 = 706.5 \, \text{cm}^3 \]

Step 3: Hemispherical Volume Calculation:

The volume of a hemisphere is calculated using \( V_{\text{hemisphere}} = \frac{2}{3} \pi r^3 \), where \( r \) is the radius. With a radius \( r = 5 \) cm, the volume of the hemispherical portion is: \[ V_{\text{hemisphere}} = \frac{2}{3} \times 3.14 \times (5)^3 = \frac{2}{3} \times 3.14 \times 125 = \frac{2}{3} \times 392.5 = 261.67 \, \text{cm}^3 \]

Step 4: Total Capacity Calculation:

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

Conclusion:

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