To compare the surface areas of the two cylinders, we need to calculate the surface area of Cylinder A using the required formulas.
Surface Area of a Cylinder = 2πr² + 2πrh
where:
- r = radius
- h = height
Let's denote the values for Cylinder A:
- Radius of Cylinder A = r_A inches
- Height of Cylinder A = h_A inches
Now, we follow through the steps given in the question.
Step 1: Area of Each Circular Face of Cylinder A
Area of Each Circular Face: \[ \text{Area}_{\text{face}} = \pi r_A^2 = 3.14 \times r_A^2 \]
Step 2: Total Area of the Circular Faces of Cylinder A
Total Area of the Circular Faces: \[ \text{Total Area}{\text{faces}} = 2 \times \text{Area}{\text{face}} = 2 \times (3.14 \times r_A^2) = 6.28 \times r_A^2 \]
Step 3: Circumference of Cylinder A
Circumference: \[ \text{Circumference} = 2\pi r_A = 2 \times 3.14 \times r_A = 6.28 \times r_A \]
Step 4: Area of the Lateral Face of Cylinder A
Area of the Lateral Face: \[ \text{Area}_{\text{lateral}} = \text{Circumference} \times h_A = (6.28 \times r_A) \times h_A = 6.28 \times r_A \times h_A \]
Step 5: Total Surface Area of Cylinder A
Total Surface Area: \[ \text{Total Surface Area} = \text{Total Area}{\text{faces}} + \text{Area}{\text{lateral}} \] \[ = (6.28 \times r_A^2) + (6.28 \times r_A \times h_A) \] \[ = 6.28 \times (r_A^2 + r_A \times h_A) \]
Now we compare Cylinder A and Cylinder B.
Given that the surface area of Cylinder B is 251.2 square inches, we need to solve \( 6.28 \times (r_A^2 + r_A \times h_A) = 251.2 \).
To determine if the student is correct, we need to evaluate this expression with actual values for \( r_A \) and \( h_A \). However, we'll need specific values for \( r_A \) and \( h_A \) to complete the calculation.
Let's say we stick with a hypothetical radius and height for Cylinder A, for example:
- Radius of Cylinder A = 5 inches
- Height of Cylinder A = 10 inches
Then we compute:
-
Area of Each Circular Face: \[ \text{Area}_{\text{face}} = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ square inches} \]
-
Total Area of the Circular Faces: \[ \text{Total Area}_{\text{faces}} = 2 \times 78.5 = 157 \text{ square inches} \]
-
Circumference of Cylinder A: \[ \text{Circumference} = 6.28 \times 5 = 31.4 \text{ inches} \]
-
Area of the Lateral Face: \[ \text{Area}_{\text{lateral}} = 31.4 \times 10 = 314 \text{ square inches} \]
-
Total Surface Area of Cylinder A: \[ \text{Total Surface Area} = 157 + 314 = 471 \text{ square inches} \]
Now, we can finalize the comparison:
- The surface area of Cylinder A: 471 square inches
- The surface area of Cylinder B: 251.2 square inches
Question 2 Answer: The surface area of Cylinder A is greater than the surface area of Cylinder B.