What would the radius and height be of a 3-dimensional cylinder that was formed by rotating a square, with 3 inch sides, around the y-axis?
Where was the y-axis located as the square rotated around it?
length of a diagonal in the square is
√(3^2 + 3^2) = √18 = 3√2
case 1, the y-axis runs down the centre of the square, so the
radius is (3/2)√2
volume = π r^2 h = π(9/2)(3) inches^3 = 13.5π inches^3 or appr 42.4 inches^3
case 2. the square is rotated with the y-axis along one of the edges,
then the radius would be 3√2
What would be the volume of the cylinder in that case?
surely for case 1 the radius is just 3/2 if you want a cylinder and not two cones joined at their base.
To find the radius and height of a 3-dimensional cylinder formed by rotating a square around the y-axis, let's break down the problem:
The square has sides of 3 inches. When rotated around the y-axis, the square forms the curved surface of the cylinder. To determine the radius of the cylinder, we need to find the distance from the y-axis, which represents the center of the cylinder, to the outer boundary of the curved surface.
The center of the square coincides with the y-axis, so the distance from the center to the outer boundary of the cylinder is exactly half of one of the square's sides. Since the square has sides of 3 inches, the radius of the cylinder is 3 inches ÷ 2 = 1.5 inches.
Now, to find the height of the cylinder, we need to determine the length of the square, which will be equal to the circumference of the cylinder. For a circle, the circumference is calculated using the formula: circumference = 2πr, where r is the radius.
Applying this to our cylinder, the circumference is 2π(1.5 inches) = 3π inches. Since the circumference of the cylinder represents the length of the square, the height of the cylinder is equal to this length. Therefore, the height of the cylinder is 3π inches.
To summarize:
- The radius of the cylinder is 1.5 inches.
- The height of the cylinder is 3π inches.