determine the diameter of closed cylindrical tank in meters if the volume is 11.3m^3 to get min. surface area.

you know that

v = π r^2 h, so h = v/(πr^2)

the area is

a = 2πr(r+h) = 2πr(r + 11.3/(πr^2))
= 2πr^2 + 22.6/r

for minimum area, da/dr=0, so
da/dr = 4πr - 22.6/r^2
= (4πr^3-22.6)/r^2

da/dr=0 when 4r^3-22.6=0, or r=1.216

The diameter d = 2r.

Why did the tank go on a diet? Because it wanted to minimize its surface area!

To determine the diameter of the closed cylindrical tank with a volume of 11.3m³ to minimize the surface area, we can use a mathematical formula.

The volume of a cylinder is given by the formula V = πr²h, where V is the volume, r is the radius, and h is the height. Since the tank is closed, we can assume the height is equal to the diameter (h = 2r).

In this case, we have V = 11.3m³ and we want to minimize the surface area, which is given by the formula A = 2πrh + πr².

To minimize the surface area, we need to take the derivative of A with respect to r and set it equal to zero. Then solve for r.

But hey, enough with the boring math talk! Let me tell you a joke instead:

Why did the clown bring a ladder to the party?
Because he wanted to climb up the social "ladder"!

To determine the diameter of the closed cylindrical tank that will result in the minimum surface area for a given volume, we need to use calculus. The surface area of a closed cylinder can be defined as:

A = 2πrh + πr^2

Where A is the surface area, r is the radius, and h is the height of the cylinder.

Given the volume V = 11.3 m^3, we have the following equation:

V = πr^2h

To find the minimum surface area, we need to minimize the surface area function A with respect to radius r. To do this, we need to find the value of the radius that makes the derivative of A with respect to r equal to zero.

Step 1: Express the surface area function A in terms of h:

V = πr^2h [Solve for h]

h = V / (πr^2)

Substitute this expression for h in the surface area function:

A = 2πr(V / (πr^2)) + πr^2
A = 2V / r + πr^2

Step 2: Differentiate the surface area function A with respect to radius r:

dA/dr = -2V / r^2 + 2πr

Step 3: Set the derivative equal to zero and solve for r:

-2V / r^2 + 2πr = 0
-2V + 2πr^3 = 0
r^3 = V / π
r = (V / π)^(1/3)

Step 4: Calculate the diameter using the radius:

Diameter = 2r
Diameter = 2(V / π)^(1/3)

Let's calculate the diameter using V = 11.3 m^3:

Diameter = 2(11.3 / π)^(1/3)
Diameter ≈ 2(3.6004)^(1/3)
Diameter ≈ 2(1.668)
Diameter ≈ 3.336 meters

Therefore, the diameter of the closed cylindrical tank that will result in the minimum surface area, given a volume of 11.3 m^3, is approximately 3.336 meters.

To find the minimum surface area of a closed cylindrical tank with a given volume, follow these steps:

Step 1: Understand the problem
We are given the volume of the cylinder, which is 11.3 m^3, and we need to find the diameter that minimizes the surface area.

Step 2: Identify the formula
The formula for the volume V of a cylinder is V = πr^2h, where r is the radius and h is the height. The formula for the surface area S of a cylinder is S = 2πr^2 + 2πrh.

Step 3: Solve for height
Since we are given the volume, we can rearrange the volume formula to solve for the height h:
V = πr^2h => h = V / (πr^2)

Step 4: Substitute the value of h
Substitute the value of h in the surface area formula:
S = 2πr^2 + 2πr(V / (πr^2))
Simplify:
S = 2πr^2 + 2V / r

Step 5: Find the derivative
To find the value of r that minimizes the surface area, we need to find the derivative of the surface area formula with respect to r:
dS/dr = 4πr - 2V / r^2

Step 6: Set the derivative equal to zero
Set the derivative equal to zero and solve for r:
4πr - 2V / r^2 = 0
4πr^3 - 2V = 0
r^3 = V / (2π)
r = (V / (2π))^(1/3)

Step 7: Find the diameter
The diameter D is twice the value of the radius:
D = 2r

Step 8: Substitute the given volume
Now, substitute the volume given in the question:
D = 2((11.3 / (2π))^(1/3))

Using a calculator, you can find the value of D, which would be the diameter of the closed cylindrical tank in meters that minimizes the surface area.