Answer:
Formula for the volume of the cylinder is given by;
![V = \pi r^2h](https://tex.z-dn.net/?f=V%20%3D%20%5Cpi%20r%5E2h)
where r is the radius of the cylinder and h be the height of the cylinder.
As per the statement:
Volume of cylinder(V) = 364.5 cubic inches.
You will cut its bottom from a square of metal and form its curved side by bending a rectangular sheet of metal.
Diameter of a circle = 2r
Then, side of the square (s) = 2r
Area of a square =
square inches.
Since, Volume of the cylinder(V) = ![\pi r^2h](https://tex.z-dn.net/?f=%5Cpi%20r%5E2h)
then;
![h = \frac{V}{\pi r^2}](https://tex.z-dn.net/?f=h%20%3D%20%5Cfrac%7BV%7D%7B%5Cpi%20r%5E2%7D)
Substitute the value of V then we have
![h = \frac{364.5}{\pi r^2}](https://tex.z-dn.net/?f=h%20%3D%20%5Cfrac%7B364.5%7D%7B%5Cpi%20r%5E2%7D)
To find the total amount of material required for the square and the rectangle in terms of r.
![\text{Total amount of material} = \text{area of the square which used to cut the bottom} + \text{curved surface of cylinder}](https://tex.z-dn.net/?f=%5Ctext%7BTotal%20amount%20of%20material%7D%20%3D%20%5Ctext%7Barea%20of%20the%20square%20which%20used%20to%20cut%20the%20bottom%7D%20%2B%20%5Ctext%7Bcurved%20surface%20of%20cylinder%7D)
Curved surface area of cylinder = ![2 \pi rh](https://tex.z-dn.net/?f=2%20%5Cpi%20rh)
Substitute the value of h we have;
Curved surface area of cylinder = ![2 \pi r \cdot \frac{364.5}{\pi r^2}](https://tex.z-dn.net/?f=2%20%5Cpi%20r%20%5Ccdot%20%5Cfrac%7B364.5%7D%7B%5Cpi%20r%5E2%7D)
=
square inches.
then;
![\text{Total amount of material} = \frac{729}{r}+4r^2](https://tex.z-dn.net/?f=%5Ctext%7BTotal%20amount%20of%20material%7D%20%3D%20%5Cfrac%7B729%7D%7Br%7D%2B4r%5E2)
=
square inches
Therefore, the total amount of material required for the square and the rectangle in terms of r is,
square inches