Make a net for the cylinder. then find the surface area of the cylinder. round your answer to the nearest tenth 4in 2in.__
The Correct Answer and Explanation is :
To find the surface area of a cylinder, we first need to create a net of the cylinder and then use that to calculate the surface area. A net is a two-dimensional representation that can be folded to form the three-dimensional shape of the cylinder.
Step 1: Create a Net for the Cylinder
A cylinder consists of two circular bases and one rectangular lateral surface. The net of a cylinder can be visualized as follows:
- Two Circles: Each base of the cylinder is represented by a circle. The radius of the circles is equal to the radius of the cylinder.
- One Rectangle: The lateral surface area is represented by a rectangle. The width of the rectangle is equal to the circumference of the base, and the height is equal to the height of the cylinder.
Given Dimensions:
- Height (h) = 4 inches
- Radius (r) = 2 inches
Step 2: Surface Area Calculation
The surface area ( S ) of a cylinder is given by the formula:
[
S = 2\pi r^2 + 2\pi rh
]
Where:
- ( 2\pi r^2 ) is the area of the two circular bases.
- ( 2\pi rh ) is the area of the rectangular lateral surface.
Step 2.1: Calculate the Area of the Circular Bases
[
\text{Area of bases} = 2\pi r^2 = 2\pi (2)^2 = 2\pi (4) = 8\pi
]
Step 2.2: Calculate the Area of the Lateral Surface
[
\text{Lateral surface area} = 2\pi rh = 2\pi (2)(4) = 16\pi
]
Step 2.3: Total Surface Area
[
S = 8\pi + 16\pi = 24\pi
]
Step 3: Numerical Calculation
To find the numerical value of the surface area, we use ( \pi \approx 3.14 ):
[
S \approx 24 \times 3.14 \approx 75.36
]
Step 4: Round the Result
Rounding ( 75.36 ) to the nearest tenth gives:
[
\text{Surface Area} \approx 75.4 \text{ square inches}
]
Conclusion
The surface area of the cylinder with a height of 4 inches and a radius of 2 inches is approximately 75.4 square inches. This calculation showcases how geometric shapes can be decomposed into simpler forms (the net), making it easier to compute their areas using known formulas. Understanding these principles is essential in various fields, including engineering, architecture, and manufacturing, where precise measurements are critical.