An illustration shows a rectangle of unknown height with a circle of radius 3 feet attached to its top left and a circle of radius 3 feet attached to its bottom right.
If the surface area of the net diagram of the right circular cylinder is 395.64 feet^2 , find the height of the net diagram. Use 3.14 for π .
Let's try a different approach:
Given:
Area of the net diagram = 395.64 ft^2
Radius of each circle = 3 ft
π = 3.14
Let the height of the rectangle be h ft.
The lateral surface area of the right circular cylinder is the area of the rectangle, which is equal to the height*h*2πr = 2πrh, where r is the radius.
The total surface area of the net diagram is the sum of the lateral surface area and the areas of the two circles.
Total surface area = 2πrh + πr^2 + πr^2
395.64 = 2 * 3.14 * h * 3 + 3.14 * 3^2 + 3.14 * 3^2
395.64 = 18.84h + 28.26 + 28.26
395.64 = 18.84h + 56.52
18.84h = 395.64 - 56.52
18.84h = 339.12
h = 339.12 / 18.84
h ≈ 18 ft
Therefore, the correct height of the net diagram is approximately 18 feet.
To find the surface area of the right circular cylinder, we need to consider three surfaces involved: the lateral surface area, the top circular surface, and the bottom circular surface.
Given:
Length of the log (height of the cylinder), h = 20 feet
Radius of the cylinder, r = 1.5 feet
π = 3.14
1. Let's calculate the lateral surface area of the cylinder:
Lateral surface area = 2πrh
= 2 * 3.14 * 1.5 * 20
= 188.4 square feet
2. The area of each circular surface (top and bottom) is given by:
Circular surface area = πr^2
= 3.14 * (1.5)^2
= 3.14 * 2.25
= 7.065 square feet
3. The total surface area of the right circular cylinder is the sum of the lateral surface area and the areas of the two circular surfaces:
Total surface area = Lateral surface area + 2 * Circular surface area
= 188.4 + 2 * 7.065
= 188.4 + 14.13
= 202.53 square feet
Therefore, the surface area of the log-shaped right circular cylinder is 202.53 square feet.
Let's denote the height of the rectangle as h.
The net diagram consists of 3 parts:
- The rectangle: its area is h * h = h^2.
- The circle on the top left: its area is π * r^2 = 3.14 * 3^2 = 28.26.
- The circle on the bottom right: its area is π * r^2 = 3.14 * 3^2 = 28.26.
The total surface area of the net diagram is the sum of these three parts:
h^2 + 28.26 + 28.26 = 395.64
h^2 + 56.52 = 395.64
h^2 = 339.12
h ≈ 18.41 feet
Therefore, the height of the net diagram is approximately 18.41 feet.
Wrong, choose a different answer.
My apologies for the mistake. Let's correct it.
The total surface area of the net diagram is given by the sum of the areas of the rectangle and the two circles.
Given:
- Area of the rectangle = h * h = h^2
- Area of each circle = π * r^2 = 3.14 * 3^2 = 28.26
Total surface area = h^2 + 28.26 + 28.26 = 395.64
h^2 + 56.52 = 395.64
h^2 = 339.12
h = √339.12
h ≈ 18.41 feet
Therefore, the correct height of the net diagram is approximately 18.41 feet.
No. That's wrong.
Let's try a different approach:
The net diagram of the right circular cylinder consists of the lateral surface area of the cylinder, which is the rectangle, and the areas of the two circles (top and bottom).
The lateral surface area of a cylinder is given by the formula: Lateral surface area = 2πrh, where r is the radius of the cylinder and h is the height.
In our case, the radius of the circles is 3 feet. So, the lateral surface area of the cylinder, which is the area of the rectangle, is equal to: 2π * 3 * h = 6πh.
The area of each circle is πr^2 = π * 3^2 = 9π.
Therefore, the total surface area of the net diagram is the sum of the lateral surface area and the areas of the two circles: 6πh + 9π + 9π = 395.64
Simplify: 6πh + 27π = 395.64
6πh = 368.64
h = 368.64 / (6π)
h = 368.64 / (6 * 3.14)
h ≈ 19.71 feet
Therefore, the correct height of the net diagram is approximately 19.71 feet.