Question

A student says the two cylinders below have the same surface area. Explain the student’s error. (Use 3.14 for π.)

1. width 3 inches length 10 inches

2 width 10 inches length 3 inches

Answer 1

[AI-generated answer — Bot GPT 3.5]

The student's error lies in assuming that the surface area of a cylinder is solely determined by the dimensions of the base. In reality, the surface area of a cylinder is determined by the formula:

Surface Area = 2πrh + 2πr^2

where r is the radius of the cylinder's base and h is the height of the cylinder.

For cylinder 1:
radius = 3/2 = 1.5 inches
height = 10 inches

Surface Area = 2(3.14)(1.5)(10) + 2(3.14)(1.5)^2 = 94.2 + 14.13 = 108.33 square inches

For cylinder 2:
radius = 10/2 = 5 inches
height = 3 inches

Surface Area = 2(3.14)(5)(3) + 2(3.14)(5)^2 = 94.2 + 157 = 251.2 square inches

Therefore, the student's error is in assuming that the two cylinders have the same surface area, when in fact they do not. The second cylinder has a much larger surface area than the first cylinder.

Answer 2

Determine the surface area of the triangular prism. (1 point) Responses 110 square centimeters 110 square centimeters 132 square centimeters 132 square centimeters 144 square centimeters 144 square centimeters 330 square centimeters 330 square centimeters

11x 4x 5x 3x

Answer 3

[AI-generated answer — Bot GPT 3.5]

To determine the surface area of a triangular prism, we need to calculate the areas of all the faces and then add them together.

The formula for the surface area of a triangular prism is:
Surface Area = 2 * base area + lateral surface area

Given that the base is a triangle with sides of 11x, 4x, and 5x, we first need to find the area of the base triangle.

Let's calculate the area of the base triangle using Heron's formula:

Semi-perimeter = (11x + 4x + 5x) / 2 = 10x
Area = √(10x(10x - 11x)(10x - 4x)(10x - 5x))
Area = √(10x * 1x * 6x * 5x)
Area = √(300x^2)
Area = 10√3x

Now, let's find the lateral surface area of the prism:
Lateral surface area = perimeter of the base * height
Lateral surface area = (11x + 4x + 5x) * 3x
Lateral surface area = 20x * 3x = 60x^2

Now, let's calculate the total surface area:
Surface Area = 2(10√3x) + 60x^2
Surface Area = 20√3x + 60x^2
Surface Area = 20x(√3 + 3x)

Therefore, the surface area of the triangular prism is 20x(√3 + 3)x or 20x(√3 + 3)x square units.