The total amount of material for the square is \(\boxed{4r^{2}}\) and for the rectangle is \(\boxed{\dfrac{729}{r}}\).
Further explanation:
The volume of the cylinder is \(364.5\text{ in}^{3}\).
The area of the square can be obtained by the formula \(\boxed{a^{2}}\) where, \(a\) is the side of the square.
The area of the rectangle is obtained by the formula is \(\boxed{l\times b}\) , where \(l\) and \(b\) are length and breadth of the rectangle respectively.
The volume of the cylinder is \(\boxed{\pi r^{2}h}\) where, \(r\) is the radius of the cylinder and \(h\) is the height of the cylinder.
The area of circle is \(\boxed{\pi r^{2}}\).
First we have to find the height of the cylinder by applying the formula of volume of cylinder,
\(\begin{aligned}\pi r^{2}h&=364.5\\h&=\dfrac{364.5}{\pi r^{2}h}\end{aligned}\)
The base of the cylinder is circle and it is obtained from the square.
Therefore, the diameter of the circle is equal to the side of square.
\(\boxed{d=2r=a}\)
Here, \(a\) is the side of square.
The area of the square is calculated as follows:
\(\begin{aligned}\text{Area of square}&=(2r)^{2}\\&=4r^{2}\end{aligned}\)
Therefore, the total amount of material require for square is \(\boxed{4r^{2}}\).
When we fold the rectangle the length of the rectangle is equal to the perimeter of the base of the cylinder.
\(\boxed{2\pi r=l}\)
The breadth of the rectangle is equal to the height of the cylinder.
\(\boxed{b=h}\)
The value of \(h\) is obtained as \(\frac{364.5}{r^{2}}\).
Therefore, the area of rectangular sheet of metal is calculated as follows:
\(\begin{aligned}\text{Area of rectangle}&=l\times b\\&=(2\pi r)\left(\dfrac{364.5}\pi {r^{2}}\right)\\&=2\left(\dfrac{3645}{10}\right)\left(\dfrac{\pi r}{\pi r^{2}}\right)\\&=\dfrac{7290}{10}\cdot \dfrac{1}{r}\\&=\dfrac{729}{r}\end{aligned}\)
Thus, the total amount of material require for rectangle is \(\boxed{\dfrac{729}{r}}\).
Learn more:
1. What is the general form of the equation of the given circle with center a? https://brainly.com/question/1506955
2. Which equation represents a circle with the same radius as the circle shown but with a center at (-1, 1)? https://brainly.com/question/1952668
Answer details:
Grade: High school
Subject: Mathematics
Chapter: Mensuration
Keywords: Rectangle, square, cylinder, amount of material, 729/r , 364.5/pir^2 ,area of circle, volume of cylinder, Material, cylinderical can, rectangular sheet.
