Calculating the Total Surface Area of a Solid: Hemisphere Mounted on a Cylinder

Introduction

The calculation of the total surface area (TSA) of a solid formed by mounting a hemisphere on a cylinder with equal radii is a classic geometric problem. This article will walk you through the detailed steps to solve such a problem, ensuring you understand each part of the calculation and its significance.

Step-by-Step Guide

Given a solid composed of a cylinder with a hemisphere mounted on it, where the total volume is 2707 cc (cubic centimeters) and the height of the cylinder is 80 cm, we seek to determine the total surface area (TSA) of the solid.

Step 1: Determine the Volume

We start by calculating the volume of the solid. The volume of a cylinder is given by V_c πr^2h, and the volume of a hemisphere is V_h (2/3)πr^3. The total volume of the solid is the sum of these two volumes:

V V_c V_h πr^2h (2/3)πr^3 2707 cc

Substituting h 80 cm, we get:

πr^2(80) (2/3)πr^3 2707

Factoring out π:

π(80r^2 (2/3)r^3) 2707

Dividing both sides by π:

80r^2 (2/3)r^3 2707/π ≈ 861

Step 2: Solve the Equation

Next, we solve the equation to find the radius r. First, we multiply the entire equation by 3 to eliminate the fraction:

240r^2 2r^3 2583

Rearranging the terms, we obtain a cubic equation:

2r^3 - 240r^2 - 2583 0

We can estimate the value of r by trying rational roots. Testing values for r:

r 7: 7^3 - 1207^2 - 1291.5 343 - 12049 - 1291.5 4431.5 (too high) r 6: 6^3 - 1206^2 - 1291.5 216 - 12036 - 1291.5 3244.5 (too high) r 5: 5^3 - 1205^2 - 1291.5 125 - 12025 - 1291.5 1833.5 (too high) r 4: 4^3 - 1204^2 - 1291.5 64 - 12016 - 1291.5 692.5 (too low) r 5.5: 5.5^3 - 1205.5^2 - 1291.5 ≈ 166.375 - 12030.25 - 1291.5 ≈ 166.375 - 12030 - 1291.5 (indicates r is between 5.5 and 6)

Using numerical methods or a calculator, we find that r ≈ 5.7 cm.

Step 3: Calculate the Total Surface Area

The total surface area (TSA) of the solid is the sum of the curved surface area (CSA) of the cylinder, the curved surface area of the hemisphere, and the area of the base of the cylinder (which is also the base of the hemisphere).

CSA_{cylinder} 2πrh 2π5.780 2π456 ≈ 2866.4 cm^2 CSA_{hemisphere} 2πr^2 2π5.7^2 ≈ 2π32.49 ≈ 204.1 cm^2 Area_{base} πr^2 ≈ π5.7^2 ≈ 32.49π ≈ 102.1 cm^2

Combining these:

TSA CSA_{cylinder} CSA_{hemisphere} Area_{base}

Therefore, the total surface area is approximately:

TSA 2866.4 204.1 102.1 ≈ 3172.6 cm^2

Conclusion

The total surface area of the solid is approximately 3172.6 cm2. This problem not only reinforces the formulas for the volumes and surface areas of cylinders and hemispheres but also highlights the importance of determining the radius from the given volumes and heights.