A student says the two cylinders below have the same surface area. Explain the student’s error. (Use 3.14 for π.) 3in 10 in 10in 3in
The student's error is that they are only comparing the areas of the bases of the two cylinders and assuming that because they are the same, the surface area of the entire cylinders must also be the same. However, surface area of a cylinder is not just the area of the bases, but also includes the lateral surface area.
To calculate the surface area of a cylinder, you need to find the area of the two bases (which are circles) and the lateral surface area (which is the curved surface around the sides of the cylinder).
For the cylinder with a radius of 3 inches and height of 10 inches:
- Base area = πr^2 = 3.14 * 3^2 = 28.26 square inches
- Lateral surface area = 2πrh = 2 * 3.14 * 3 * 10 = 188.4 square inches
- Total surface area = 2(base area) + lateral surface area = 2(28.26) + 188.4 = 244.92 square inches
For the cylinder with a radius of 10 inches and height of 3 inches:
- Base area = πr^2 = 3.14 * 10^2 = 314 square inches
- Lateral surface area = 2πrh = 2 * 3.14 * 10 * 3 = 188.4 square inches
- Total surface area = 2(base area) + lateral surface area = 2(314) + 188.4 = 816.4 square inches
Therefore, the two cylinders do not have the same surface area. The second cylinder with a larger radius and smaller height has a much larger surface area compared to the first cylinder with a smaller radius and larger height.