A cylinder has a circumference of 16pi inches and its height is half the radius of the cylinder? What is the total surface area of the cylinder? Give answer in terms of pi.
do not know how to do this....please help
let the radius be r
2πr = 16π
r = 8
so the height is 4 inches and the radius is 8 inches.
SA = 2 circles + a rectangle
= 2πr^2 + 2πrh
= 2π(8)^2 + 2π(8)(4)
= ....
64pi
To find the total surface area of a cylinder, we need to find the sum of the areas of the two bases and the lateral surface area.
1. Let's start by finding the radius of the cylinder. The circumference of a circle is given by the formula:
circumference = 2 * pi * radius
Given that the circumference is 16pi inches, we can solve for the radius:
16pi = 2 * pi * radius
Dividing both sides of the equation by 2 * pi, we get:
16pi / (2 * pi) = radius
Simplifying, we find that the radius of the cylinder is 8 inches.
2. Given that the height of the cylinder is half the radius, we can find the height:
height = 1/2 * 8
Simplifying, we find that the height of the cylinder is 4 inches.
3. Now, let's calculate the area of each base. The area of a circle is given by the formula:
area = pi * radius^2
The radius of the base is 8 inches, so the area of each base is:
base area = pi * (8)^2
Simplifying, we find that the area of each base is 64pi square inches.
4. Finally, let's calculate the lateral surface area. The lateral surface area of a cylinder is given by the formula:
lateral surface area = 2 * pi * radius * height
The radius of the cylinder is 8 inches and the height is 4 inches, so the lateral surface area is:
lateral surface area = 2 * pi * 8 * 4
Simplifying, we find that the lateral surface area is 64pi square inches.
5. To find the total surface area, we add the areas of the two bases and the lateral surface area:
total surface area = 2 * base area + lateral surface area
total surface area = 2 * (64pi) + (64pi)
Simplifying, we find that the total surface area of the cylinder is 192pi square inches.
Therefore, the total surface area of the cylinder is 192pi square inches.
To find the total surface area of a cylinder, we need to add the areas of the two circular bases and the area of the curved surface.
Let's break down the problem step by step.
Step 1: Find the radius of the cylinder.
The circumference of a circle is given by the formula C = 2πr, where C is the circumference and r is the radius. In this case, the circumference is 16π inches. So, we can set up the equation:
16π = 2πr
Divide both sides of the equation by 2π:
r = 8 inches
Step 2: Find the height of the cylinder.
The height is given as half the radius, so:
height = 1/2 * radius
height = 1/2 * 8 inches
height = 4 inches
Step 3: Calculate the areas of the circular bases.
The area of a circle is given by the formula A = πr^2.
Since we already know the radius is 8 inches, we can calculate the area:
Base area = π * (8 inches)^2
Base area = 64π square inches
Step 4: Calculate the area of the curved surface.
The curved surface area of a cylinder is given by the formula A = 2πrh, where r is the radius and h is the height.
In this case, r = 8 inches and h = 4 inches.
Curved surface area = 2π * 8 inches * 4 inches
Curved surface area = 64π square inches
Step 5: Calculate the total surface area.
The total surface area of a cylinder is obtained by adding the areas of the two bases and the curved surface area.
Total surface area = 2(Base area) + Curved surface area
Total surface area = 2(64π square inches) + 64π square inches
Total surface area = 128π square inches + 64π square inches
Total surface area = 192π square inches
Therefore, the total surface area of the given cylinder is 192π square inches.