Find the radius and hight of cylinder with voume 64π and radius r between 1 and 5 that has smallest possible surface area. A cylinder of radius r and hight aah 2π^2+2π r h and π r ^(2) h.
V = r ^ 2 ð h
64 ð = r ^ 2 ð h Divide both sides with ð
64 = r ^ 2 h Divide both sides with r ^ 2
64 / r ^ 2 = h
h = 64 / r ^ 2
A = 2 r ^ 2 ð + 2 r ð h
A = 2 r ^ 2 ð + 2 r ð 64 / r ^ 2
A = 2 r ^ 2 ð + 128 r ð / r ^ 2
A = 2 r ^ 2 ð + 128 ð / r
d A / d r = ( 4 ð ( r ^ 3 - 32 ) ) / r ^ 2
Function has extreme value where first derivative = 0
dA / dr = 0
Solutions:
r = -1.5874 + 2.7495 i
r = -1.5874 2.7495 i
and
r = 2 2 ^( 2 / 3 ) = 3,1748
So :
r = 2 2 ^( 2 / 3 ) = 3,1748
Second derivative :
4 ð ( 64 / r ^ 3 + 1 ) =
4 ð ( 64 / 32 + 1 ) =
4 ð ( 2 + 1 ) =
4 ð 3 = 12 ð > 0
Remark : r ^ 3 = [ 2 2 ^( 2 / 3 ) ] ^ 3 = 32
If second derivative > 0
that is minimum value of function.
So :
r = 2 64 / [ 2 2 ^( 2 / 3 ) ] ^ 2
h = 64 / r ^ 2
h = h = 64 / [ 2 2 ^( 2 / 3 ) ] ^ 2
h = 4 2 ^( 2 / 3 )
ð = pi number
To find the radius and height of a cylinder with the smallest possible surface area, given its volume and a range of possible radius values, we need to consider the formulas for the volume and surface area 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, and h is the height of the 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 of the cylinder.
Given that the volume of the cylinder is 64π, we can set up the following equation:
64π = πr^2h
Simplifying the equation, we have:
64 = r^2h
To find the radius and height of the cylinder that minimizes its surface area, we need to express the surface area in terms of a single variable. We can use the volume equation to express the height, h, in terms of the radius, r:
h = 64/(r^2)
Substituting this value of h in the surface area equation, we get:
A = 2πr(64/(r^2)) + 2π(r^2)
Simplifying the equation further:
A = (128π/r) + 2πr^2
Now, we have the surface area expressed as a function of the radius, A(r). To find the minimum surface area, we need to find the critical points of this function. We can achieve this by taking the derivative of A(r) with respect to r and setting it equal to zero:
dA/dr = -128π/r^2 + 4πr = 0
Solving for r, we have:
-128π/r^2 + 4πr = 0
-128π + 4πr^3 = 0
r^3 = 32
r = ∛32
r = 2
Now that we have the radius, we can substitute it back into the volume equation to find the height:
64π = π(2^2)h
64π = 4πh
h = 64/4
h = 16
Therefore, the radius of the cylinder with the smallest possible surface area is 2, and the height is 16.