Question

A solid is in the form of a cylinder with hemispherical ends of the same radii. The total height of the solid is 20 cm and the diameter of the cylinder is 14 cm. Find the surface area of the solid.

Answer 1

Step 1: Understanding the problem:

We are given a solid that is in the form of a cylinder with hemispherical ends of the same radii. The total height of the solid is 20 cm, and the diameter of the cylinder is 14 cm. We need to find the surface area of the solid.
To find the surface area, we need to calculate the areas of the cylindrical surface and the areas of the hemispherical ends.

Step 2: Breaking down the surface area calculation:

The surface area of the solid consists of: 1. The lateral surface area of the cylindrical portion. 2. The surface area of the two hemispherical ends.
The total height of the solid is 20 cm, which includes both the cylindrical part and the two hemispherical ends. Since the two hemispheres are at both ends of the cylinder, the height of the cylindrical portion is: \[ h_{\text{cylinder}} = 20 - 2r \] where \( r \) is the radius of the cylinder and hemispheres.
Given that the diameter of the cylinder is 14 cm, the radius \( r \) is: \[ r = \frac{14}{2} = 7 \, \text{cm} \] Now, calculate the height of the cylindrical portion: \[ h_{\text{cylinder}} = 20 - 2 \times 7 = 20 - 14 = 6 \, \text{cm} \]

Step 3: Surface area of the cylindrical portion:

The lateral surface area \( A_{\text{cylinder}} \) of a cylinder is given by the formula: \[ A_{\text{cylinder}} = 2\pi r h_{\text{cylinder}} \] Substitute \( r = 7 \, \text{cm} \) and \( h_{\text{cylinder}} = 6 \, \text{cm} \): \[ A_{\text{cylinder}} = 2 \pi \times 7 \times 6 = 84 \pi \, \text{cm}^2 \]

Step 4: Surface area of the hemispherical ends:

The surface area \( A_{\text{hemisphere}} \) of one hemisphere is given by: \[ A_{\text{hemisphere}} = 2\pi r^2 \] Since there are two hemispheres, the total surface area of the hemispherical ends is: \[ A_{\text{total hemispheres}} = 2 \times 2\pi r^2 = 4\pi r^2 \] Substitute \( r = 7 \, \text{cm} \): \[ A_{\text{total hemispheres}} = 4 \pi \times 7^2 = 4 \pi \times 49 = 196 \pi \, \text{cm}^2 \]

Step 5: Total surface area of the solid:

The total surface area \( A_{\text{total}} \) of the solid is the sum of the lateral surface area of the cylinder and the surface area of the hemispherical ends: \[ A_{\text{total}} = A_{\text{cylinder}} + A_{\text{total hemispheres}} = 84\pi + 196\pi = 280\pi \, \text{cm}^2 \] Using \( \pi = 3.14 \), we can approximate the total surface area: \[ A_{\text{total}} = 280 \times 3.14 = 879.2 \, \text{cm}^2 \]

Step 6: Conclusion:

The surface area of the solid is approximately \( 879.2 \, \text{cm}^2 \).