Determine the diameter of a close cylindrical tank having a volume of 11.3 cubic meter that would give the smallest surface area
1/4 d^2 h = 11.3
h = 45.2/d^2
a = π/2 d^2 + πdh = π/2 d^2 + 45.2π/d
da/dd = πd - 45.2π/d^2
da/dd=0 at d=3.56
To find the diameter of a closed cylindrical tank that would give the smallest surface area for a given volume, we can use calculus.
Let's start with the formula for the volume of a cylinder: V = πr^2h, where V is the volume, r is the radius, and h is the height of the cylinder.
In this case, we are given the volume V = 11.3 cubic meters. So we have:
11.3 = πr^2h
Next, we need to express the surface area of the cylinder in terms of r (radius) only. The surface area (A) of the cylinder consists of two parts: the area of the top and bottom circles (2πr^2) and the area of the curved side (2πrh).
A = 2πr^2 + 2πrh
Now, let's solve the equation for h in terms of r:
11.3 = πr^2h
h = 11.3 / (πr^2)
Substituting this value of h in the equation for surface area, we get:
A = 2πr^2 + 2πr(11.3 / (πr^2))
A = 2πr^2 + (22.6 / r)
To find the minimum surface area, we need to find the critical points by taking the derivative of A with respect to r and set it equal to zero:
dA/dr = 4πr - (22.6 / r^2) = 0
Simplifying the equation, we get:
4πr = 22.6 / r^2
Multiply both sides by r^2:
4πr^3 = 22.6
Divide by 4π:
r^3 = 22.6 / (4π)
Take the cube root of both sides to solve for r:
r = (∛(22.6) / (∛(4π))
Now, we have found the radius of the cylinder that gives the minimum surface area. To find the diameter, simply multiply the radius by 2:
diameter = 2 * r
So, to determine the diameter of the closed cylindrical tank that would give the smallest surface area for a volume of 11.3 cubic meters, you would need to calculate (∛(22.6) / (∛(4π))) and multiply it by 2.