Find the minimum amount of sheet that can be made into a closed clinder having a volume of 120 cu inches.
let the radius of the cylinder be r inches
and its height be h inches
given: V = 120 inches^3
or
π r^2 h inches^3 = 120 inches^3
h = 120/(πr^2)
Surface area
= 2πr^2 + 2πrh
= 2πr^2 + 2πr(120/πr^2)
= 2πr^2 + 240/r
d(surface area)/dr = 4πr - 240/r^2
= 0 for a min of surface area
4πr = 240/r^3
r^3 = 60/π
r = (60/π)^(1/3) = appr 2.673 inches
sub that back into
2πr^2 + 240/r to find the min surface area
A = 2 pi r h + 2 *pi r^2
V = 120 = pi r^2 h
so h = 120/ (pi r^2)
A = 2 pi r 120/ (pi r^2) + 2 pi r^2
A = 240 / r + 2 pi r^2
dA/dr = -240/r^2 + 4 pi r
= 0 for max or min
4 pi r = 240/r^2
r^3 = 60 / pi
do not trust my arithmetic, working fast
To find the minimum amount of sheet that can be made into a closed cylinder, we need to first determine the dimensions of the cylinder.
The volume of a cylinder can be calculated using the formula:
Volume = π * r^2 * h
where π (pi) is a mathematical constant (approximately 3.14159), r is the radius of the base, and h is the height of the cylinder.
To find the minimum amount of sheet, we want to minimize the surface area of the cylinder. The surface area of a cylinder consists of two circular bases and a curved surface. We can calculate the surface area using the formula:
Surface Area = 2πr^2 + 2πrh
Given that the volume of the cylinder is 120 cubic inches, we can set up an equation using the volume formula:
Volume = πr^2h = 120
Now, let's solve these equations simultaneously to find the minimum amount of sheet.
1. Solve the volume equation for h:
πr^2h = 120
h = 120 / (πr^2)
2. Substitute the value of h in the surface area formula:
Surface Area = 2πr^2 + 2πr * (120 / (πr^2))
Surface Area = 2πr^2 + 240 / r
Next, we can take the derivative of the surface area equation and set it equal to zero to find the critical points:
d(Surface Area) / dr = 4πr - 240 / r^2 = 0
To solve this equation, multiply through by r^2:
4πr^3 - 240 = 0
4πr^3 = 240
Divide by 4π:
r^3 = 60 / π
Take the cube root of both sides:
r = (60 / π)^(1/3)
Now that we have the radius, we can substitute it back into the volume formula to solve for h:
h = 120 / (π * ((60 / π)^(1/3))^2)
Simplify further to get the exact values of r and h. Then, plug the values into the surface area formula to find the minimum amount of sheet required for the closed cylinder.