Answer :
To solve the problem, we need to find the lateral area and the surface area of the right cylinder.
Step 1: Determine the radius of the cylinder.
The circumference of the base of the cylinder is given as [tex]\(16 \pi \, \text{cm}\)[/tex].
The formula for the circumference of a circle is:
[tex]\[ C = 2 \pi r \][/tex]
So, we set up the equation:
[tex]\[ 16 \pi = 2 \pi r \][/tex]
To find the radius [tex]\(r\)[/tex], divide both sides by [tex]\(2 \pi\)[/tex]:
[tex]\[ r = \frac{16 \pi}{2 \pi} = 8 \, \text{cm} \][/tex]
Step 2: Determine the height of the cylinder.
We are told that the height [tex]\(h\)[/tex] is half of the radius:
[tex]\[ h = \frac{1}{2} \times 8 = 4 \, \text{cm} \][/tex]
Step 3: Calculate the lateral area of the cylinder.
The formula for the lateral area [tex]\(L\)[/tex] of a cylinder is:
[tex]\[ L = 2 \pi r h \][/tex]
Substitute the values for [tex]\(r\)[/tex] and [tex]\(h\)[/tex]:
[tex]\[ L = 2 \pi \times 8 \times 4 = 201.1 \, \text{cm}^2 \][/tex] (rounded to one decimal place)
Step 4: Calculate the surface area of the cylinder.
The surface area [tex]\(S\)[/tex] is the sum of the lateral area and the area of the two circular bases. The formula for the surface area is:
[tex]\[ S = L + 2 \pi r^2 \][/tex]
First, calculate the area of the two bases:
[tex]\[ 2 \pi r^2 = 2 \pi \times 8^2 = 128 \pi \][/tex]
So:
[tex]\[ S = 201.1 + 128 \pi = 603.2 \, \text{cm}^2 \][/tex] (rounded to one decimal place)
So, the lateral area is approximately [tex]\(201.1 \, \text{cm}^2\)[/tex] and the surface area is approximately [tex]\(603.2 \, \text{cm}^2\)[/tex].
The correct choice is:
[tex]\[ L \approx 201.1 \, \text{cm}^2 ; S \approx 603.2 \, \text{cm}^2 \][/tex]
Step 1: Determine the radius of the cylinder.
The circumference of the base of the cylinder is given as [tex]\(16 \pi \, \text{cm}\)[/tex].
The formula for the circumference of a circle is:
[tex]\[ C = 2 \pi r \][/tex]
So, we set up the equation:
[tex]\[ 16 \pi = 2 \pi r \][/tex]
To find the radius [tex]\(r\)[/tex], divide both sides by [tex]\(2 \pi\)[/tex]:
[tex]\[ r = \frac{16 \pi}{2 \pi} = 8 \, \text{cm} \][/tex]
Step 2: Determine the height of the cylinder.
We are told that the height [tex]\(h\)[/tex] is half of the radius:
[tex]\[ h = \frac{1}{2} \times 8 = 4 \, \text{cm} \][/tex]
Step 3: Calculate the lateral area of the cylinder.
The formula for the lateral area [tex]\(L\)[/tex] of a cylinder is:
[tex]\[ L = 2 \pi r h \][/tex]
Substitute the values for [tex]\(r\)[/tex] and [tex]\(h\)[/tex]:
[tex]\[ L = 2 \pi \times 8 \times 4 = 201.1 \, \text{cm}^2 \][/tex] (rounded to one decimal place)
Step 4: Calculate the surface area of the cylinder.
The surface area [tex]\(S\)[/tex] is the sum of the lateral area and the area of the two circular bases. The formula for the surface area is:
[tex]\[ S = L + 2 \pi r^2 \][/tex]
First, calculate the area of the two bases:
[tex]\[ 2 \pi r^2 = 2 \pi \times 8^2 = 128 \pi \][/tex]
So:
[tex]\[ S = 201.1 + 128 \pi = 603.2 \, \text{cm}^2 \][/tex] (rounded to one decimal place)
So, the lateral area is approximately [tex]\(201.1 \, \text{cm}^2\)[/tex] and the surface area is approximately [tex]\(603.2 \, \text{cm}^2\)[/tex].
The correct choice is:
[tex]\[ L \approx 201.1 \, \text{cm}^2 ; S \approx 603.2 \, \text{cm}^2 \][/tex]
