A cylindrical can is made from tin.If it can be contain 1000 m^3 liquid inside it then what is the parameter of design if we are oblige use the minimum amount of tin.

To find the design parameters that minimize the amount of tin used for the cylindrical can, we need to consider the formula for the volume of a cylinder as well as its surface area.

1. Volume of a cylinder:
The volume of a cylinder is given by the formula V = πr^2h, where V is the volume, r is the radius of the base, and h is the height or length.

2. Surface area of a cylinder:
The surface area of a cylinder is given by the formula A = 2πrh + 2πr^2, where A is the surface area, r is the radius, and h is the height or length.

Now, let's find the design parameters for the cylinder that minimize the amount of tin used:

Given:
Volume of the liquid = 1000 m^3

1. Volume optimization:
To minimize the amount of tin used, we need to minimize the surface area of the cylinder, while maintaining the volume.

Since the volume is fixed at 1000 m^3, we can rearrange the volume formula to solve for the height in terms of the radius:
V = πr^2h
h = V / (πr^2)
h = 1000 / (πr^2)

2. Surface area optimization:
Now, using the formula for the surface area of a cylinder, we can substitute the derived value of h to get the surface area in terms of only the radius:
A = 2πrh + 2πr^2
A = 2πr * (1000 / (πr^2)) + 2πr^2
A = 2000 / r + 2πr^2

To minimize the surface area, we need to find the value of r that minimizes the surface area. We can do this by finding the value of r where the derivative of the surface area equation is equal to zero. The derivative is calculated as follows:

dA/dr = -2000 / r^2 + 4πr

Setting the derivative equal to zero:
-2000 / r^2 + 4πr = 0

Simplifying the equation:
-2000 + 4πr^3 = 0
4πr^3 = 2000
r^3 = 2000 / (4π)
r^3 = 500 / π
r ≈ 6.348 m

Now that we have the radius, we can substitute it back into the equation for h to find the corresponding value:
h ≈ 1000 / (π * (6.348)^2)

Therefore, the design parameters that minimize the amount of tin used are approximately:
Radius ≈ 6.348 m
Height ≈ 25.191 m (calculated using the derived radius value)