A cylindrical can is made from tin.If it can be contain liquid inside it then what is the parameter of design if we are oblige use the minimum amount of tin.
assuming a constant volume v, we have
v = pi r^2 h
so, h = v/(pi r^2)
the surface area is
a = 2pi r(r+h)
= 2pi r(r + v/(pi r^2))
= 2 pi r^2 + 2v/r
da/dr = 4pi r - 2v/r^2
= 2(2pi r^3 - v)/r^2
we want da/dr=0 for max/min area, so
r = ∛(v/(2pi))
h = v/(pi r^2) = ∛(4v/pi)
but, is this min or max area? Check a'' to be sure it's a minimum
To design a cylindrical can that minimizes the amount of tin used while still being able to hold liquid, we need to determine the parameter that will achieve this. The parameter we are looking for in this case is the ratio of the can's height to its radius, known as the aspect ratio.
Let's go through the steps to determine this parameter:
1. Consider the formula for the surface area of a cylinder:
Surface Area = 2πr² + 2πrh
In this case, we want to minimize the amount of tin used, so we need to minimize the surface area.
2. We have two variables: the height (h) and the radius (r). Since we want to minimize the surface area, we need to express the surface area equation in terms of a single variable.
3. We can find the volume of the can using the formula:
Volume = πr²h
Since we need to hold a specific amount of liquid, the volume of the can is fixed. Let's say the desired volume is V.
4. Rearrange the volume equation to express height in terms of the desired volume and the radius:
h = V / (πr²)
5. Substitute the value of h in the surface area equation:
Surface Area = 2πr² + 2πr(V / (πr²))
= 2πr² + 2V/r
6. We now have the surface area expression in terms of a single variable, r. To find the minimum surface area and thus minimize the amount of tin used, we can take the derivative of the surface area equation with respect to r and set it equal to zero.
7. Differentiate the surface area equation with respect to r:
d(Surface Area)/dr = 4πr - 2V/r²
8. Set the derivative equal to zero and solve for r:
4πr - 2V/r² = 0
4πr = 2V/r²
2πr³ = V
r³ = V / (2π)
9. Solve for r by taking the cube root of both sides of the equation:
r = (V / (2π))^(1/3)
10. Now that we have the value of r, we can substitute it back into the equation for h that we derived earlier to find the corresponding value of h:
h = V / (πr²)
11. Finally, we can calculate the aspect ratio by dividing the height by the radius:
Aspect ratio = h / r
By following these steps, we can determine the parameter of the design, which is the aspect ratio, in order to use the minimum amount of tin while ensuring that the cylindrical can can hold a specific volume of liquid.