a cylindrical oil tank of radius "r" and volume "V" is closed at both and express the total surface area "a" in term of "V" and "R" if the volume is pi/4 * m ^ 3,find the volume of "r" which makes the area "A" the minimum
To find the surface area "A" in terms of volume "V" and radius "r", we need to consider the formula for the surface area of a closed cylindrical oil tank.
The surface area of a closed cylindrical tank can be divided into three parts: the top and bottom circles, and the curved side.
1. The area of each circular top/bottom can be calculated using the formula: A_top_bottom = π*r^2. Since there are two top/bottom circles, the total area of the top and bottom circles is: 2π*r^2.
2. The curved side area of the cylinder can be calculated using the formula: A_side = 2π*r*h, where "h" is the height of the cylinder. As the cylinder is closed at both ends, the height is given by: h = (V / (π*r^2)).
Therefore, the curved side area is: A_side = 2π*r*(V / (π*r^2)) = 2V/r.
To find the total surface area "A", we sum up the areas of the top/bottom circles and the curved side: A = 2π*r^2 + 2V/r.
To find the value of "r" that minimizes the surface area "A", we can take the derivative of "A" with respect to "r" and set it equal to zero:
dA/dr = 4π*r - 2V/r^2 = 0
Simplifying, we have:
4π*r = 2V/r^2
2π*r^3 = V
r = (V / (2π))^(1/3)
Thus, the volume "V" and the radius "r" are related by the formula:
V = 2π*r^3
To find the volume "V" for a given radius "r" that minimizes the surface area "A", substitute the value of "r" above into the formula for volume:
V = 2π*((V / (2π))^(1/3))^3
Simplifying,
V = 2π*(V^(1/3))^3
V = 2π*V
This equation implies that "V" can be any positive value, which means there is no specific volume of "r" that makes the surface area "A" minimum.