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

Problem Statement:

A solid composed of a cylinder with hemispherical ends of identical radii has a total height of 20 cm and a cylinder diameter of 14 cm. The objective is to determine the solid's surface area.
This involves calculating the surface area of the cylindrical section and the combined surface area of the two hemispherical ends.

Surface Area Components:

The total surface area comprises:1. The lateral surface area of the cylindrical portion.2. The surface area of the two hemispherical ends.
The total height (20 cm) accounts for the cylinder and both hemispherical caps. The height of the cylindrical component is derived as:\[h_{\text{cylinder}} = 20 - 2r\]where \( r \) represents the radius of both the cylinder and the hemispheres.
Given a cylinder diameter of 14 cm, the radius \( r \) is calculated as:\[r = \frac{14}{2} = 7 \, \text{cm}\]The height of the cylindrical portion is then:\[h_{\text{cylinder}} = 20 - 2 \times 7 = 20 - 14 = 6 \, \text{cm}\]

Cylindrical Surface Area Calculation:

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

Hemispherical Ends Surface Area Calculation:

The surface area \( A_{\text{hemisphere}} \) of a single hemisphere is:\[A_{\text{hemisphere}} = 2\pi r^2\]For two hemispheres, the total surface area is:\[A_{\text{total hemispheres}} = 2 \times 2\pi r^2 = 4\pi r^2\]Substituting \( r = 7 \, \text{cm} \):\[A_{\text{total hemispheres}} = 4 \pi \times 7^2 = 4 \pi \times 49 = 196 \pi \, \text{cm}^2\]

Total Surface Area of the Solid:

The total surface area \( A_{\text{total}} \) is the sum of the cylindrical lateral surface area and the total 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 the approximation \( \pi = 3.14 \), the total surface area is approximately:\[A_{\text{total}} = 280 \times 3.14 = 879.2 \, \text{cm}^2\]

Final Result:

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