Find the dimensions (in units) of the closed cylinder with volume
V = 16π
units3 that has the least amount of surface area.
radius
units
height
units
To find the dimensions of the closed cylinder with the least amount of surface area, we can use the formula for the surface area of a closed cylinder:
Surface Area = 2πr² + 2πrh
Given that the volume of the cylinder is V = 16π units³, we can use the formula for the volume of a cylinder:
Volume = πr²h
We need to minimize the surface area, so we need to find the values of r (radius) and h (height) that satisfy the volume equation and minimize the surface area equation.
Substituting the given volume into the volume equation:
16π = πr²h
Dividing both sides by π:
16 = r²h
Now we can express either r or h in terms of the other variable:
h = 16/r²
Substituting this expression for h into the surface area equation:
Surface Area = 2πr² + 2πr(16/r²)
Simplifying:
Surface Area = 2πr² + 32π/r
To find the minimum surface area, we need to find the critical points of the surface area equation. To do this, we take the derivative of the surface area equation with respect to r and set it equal to zero.
d(Surface Area)/dr = 4πr - 32π/r² = 0
To solve for r, we can multiply both sides of the equation by r²:
4πr³ - 32π = 0
Dividing by 4π and factoring out π:
r³ - 8 = 0
(r - 2)(r² + 2r + 4) = 0
Since the surface area cannot be negative, we ignore the positive value of r and consider only the value r = 2.
Now substitute this value of r into the equation for h:
h = 16/r² = 16/2² = 16/4 = 4 units
So, the dimensions of the closed cylinder with the least amount of surface area are:
Radius = 2 units
Height = 4 units
To find the dimensions of the closed cylinder with the least surface area while having a given volume, you need to optimize the ratio of the height to the radius. The surface area of a closed cylinder is given by the formula:
Surface Area = 2πr^2 + 2πrh
Let's proceed step by step to find the dimensions:
1. Start with the volume of the cylinder, given as V = 16π units^3.
We can set up the volume equation as follows:
V = πr^2h
2. Substitute the given value of V = 16π into the equation:
16π = πr^2h
3. Simplify the equation by canceling out the π on both sides:
16 = r^2h
4. Now, we need to express the surface area in terms of a single variable to differentiate it and find the minimum. We can solve the volume equation for h:
h = (16/r^2)
5. Substitute the value of h in terms of r into the surface area equation:
Surface Area = 2πr^2 + 2πr(16/r^2)
Surface Area = 2πr^2 + 32π/r
6. Simplify the equation:
Surface Area = 2πr^2 + 32π/r
Now, the goal is to find the value of r that minimizes the surface area. We can do this by taking the derivative of the surface area with respect to r and setting it equal to zero:
d(Surface Area)/dr = 4πr - 32π/r^2 = 0
Solving this equation will give us the value of r that minimizes the surface area. Let's proceed with differentiating:
4πr - 32π/r^2 = 0
Multiply through by r^2:
4πr^3 - 32π = 0
Divide through by 4π:
r^3 - 8 = 0
Solving for r gives us:
r = ∛8
r ≈ 2.0 units
Now that we have the value of r, we can substitute it back into the equation for h:
h = (16/r^2)
h = (16/(2^2))
h = (16/4)
h = 4 units
Therefore, the dimensions of the closed cylinder with the least surface area and a volume of 16π units^3 are:
radius = 2.0 units
height = 4 units
If πr^2h=V then h=V/(πr^2)
the area is thus
a = 2πr^2+2πrh = 2πr^2+2πrV/(πr^2)
= 2πr^2 + 2V/r
minimize that