A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 2 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area. (Round your answer to three decimal places.)
if the cylinder is of radius r and height h, the volume v is
v = 4/3 pi r^3 + pi r^2 h
so, h = (v - 4/3 pi r^3)/(pi r^2)
h = (2 - 4/3 pi r^3)/(pi r^2)
the surface area a is
a = 4pi r^2 + 2pi r h
= 4pi r^2 + 2pi r (2 - 4/3 pi r^3)/(pi r^2)
= 4pi r^2 + 2/r (2 - 4/3 pi r^3)
= 4pi r^2 + 4/r - 8pi/3 r^2
= (4pi - 8pi/3) r^2 + 4/r
= 4pi/3 r^2 + 4/r
maximum area where da/dr = 0
da/dr = 8pi/3 r - 4/r^2
= (8pi/3 r^3 - 4)/r^2
da/dr=0 when r = ∛(3/2π)
As usual, check my math to verify result.
Is pi in the denominator?
yes.
what is r?
To find the radius of the cylinder that produces the minimum surface area, we need to use calculus. We can start by assigning variables to the given dimensions of the solid.
Let's assume that the radius of the cylinder is "r" and the height of the cylinder is "h". Since the height of the cylinder is not explicitly given, we will express it in terms of the radius.
The volume of the solid is given as 2 cubic centimeters. Since the solid is formed by joining two hemispheres to the ends of the cylinder, we can calculate the volume as follows:
Volume = Volume of Cylinder + 2 * Volume of Hemisphere
Volume of Cylinder = π * r^2 * h
Volume of Hemisphere = (2/3) * π * r^3
Since the volume is given as 2 cubic centimeters, we can write the equation:
2 = π * r^2 * h + 2 * (2/3) * π * r^3
Simplifying this equation:
2 = π * r^2 * h + (4/3) * π * r^3
Next, we need to express the height (h) in terms of the radius (r). Since the height is equal to the diameter of the hemisphere, which is 2r, we have:
h = 2r
Substituting this value into the equation:
2 = π * r^2 * (2r) + (4/3) * π * r^3
Simplifying further:
2 = 2πr^3 + (4/3)πr^3
Now, our equation is in terms of the radius (r) only. To find the radius that produces the minimum surface area, we need to minimize the surface area function.
The surface area of the solid is given by:
Surface Area = Area of Cylinder + 2 * Area of Hemisphere
Area of Cylinder = 2πr^2
Area of Hemisphere = 2πr^2
Surface Area = 2πr^2 + 2 * 2πr^2
Surface Area = 2πr^2 + 4πr^2
Surface Area = 6πr^2
Now, we have the surface area in terms of the radius (r).
To find the minimum surface area, we can take the derivative of the surface area function with respect to the radius (r):
d(Surface Area)/dr = d(6πr^2)/dr
d(Surface Area)/dr = 12πr
The minimum surface area occurs when the derivative is zero. So, we set the derivative equal to zero and solve for r:
12πr = 0
Dividing both sides by 12π:
r = 0
Therefore, the radius of the cylinder that produces the minimum surface area is 0.
However, this does not make sense in the context of the problem since a cylinder with zero radius would not exist. Hence, there is no minimum surface area for the given solid.
Please note that for this particular problem, the answer does not yield a valid physical solution.