Determine the height and radius of cylinder of volume 512 cm3 which has the least total surface area
To determine the height and radius of a cylinder with a volume of 512 cm³ that has the least total surface area, we need to proceed as follows:
Step 1: Understand the problem
Let's define the variables h as the height and r as the radius of the cylinder.
Step 2: Identify the relevant formulas
The volume (V) of a cylinder is given by the formula V = πr²h, and the total surface area (A) is given by the formula A = 2πrh + 2πr².
Step 3: Express the problem mathematically
We need to find the values of r and h that minimize the total surface area while satisfying the volume requirement V = 512 cm³.
Step 4: Relate the equations
From the volume equation, we can solve for h: h = V / (πr²).
Substitute this expression for h in the total surface area formula to get:
A = 2πr(V / (πr²)) + 2πr².
Step 5: Simplify the equation
Simplifying the equation further, we get:
A = 2V / r + 2πr².
Step 6: Differentiate the equation with respect to r
To find the values of r and h where the total surface area is a minimum, we differentiate the equation above with respect to r.
dA/dr = -2V/r² + 4πr.
Step 7: Set the derivative equal to zero
Set the derivative equal to zero to find the critical points:
-2V/r² + 4πr = 0.
Step 8: Solve for r
Simplify the equation further:
2πr = 2V/r².
r³ = V/π.
r³ = 512/π (since V = 512 cm³).
Step 9: Calculate r
Using a calculator, find the cube root of (512/π) to get the value of r.
Step 10: Calculate h
Use the expression for h: h = V / (πr²), and substitute the calculated value of r along with the given volume.
Step 11: Determine the lowest total surface area
Plug the values of r and h into the total surface area equation A = 2V / r + 2πr² to calculate the respective value of A.
The resulting values of r, h, and A will give you the height and radius of the cylinder with the least total surface area for the given volume.
πr^2h = 512
So, h = 512/(πr^2)
area a = 2πr^2 + 2πrh
= 2πr^2 + 2πr (512)/(πr^2)
= 2πr^2 + 1024/r
Assuming you have calculus, we see that
da/dr = 4πr - 1024/r^2
da/dr=0 when r = 4∛(4/π)
Now you can find h.