determine the diameter of closed cylindrical tank in meters if the volume is 11.3m^3 to get min. surface area
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To determine the diameter of a closed cylindrical tank that has a minimum surface area while having a volume of 11.3 m^3, we can use the concept of optimization. The surface area of a closed cylindrical tank consists of two components: the area of the circular bases and the lateral surface area.
Let's break down the problem into steps:
Step 1: Define the variables:
Let's assume that the diameter of the cylindrical tank is represented by "d" in meters.
Step 2: Formulate the equations:
The volume of a cylinder is given by the formula V = πr^2h, where r represents the radius of the cylinder's base and h is the height.
To get the minimum surface area, we need to find the dimensions that minimize the total surface area.
The lateral surface area (A_lateral) of a cylindrical tank is given by A_lateral = 2πrh.
Now, we need to express the height (h) in terms of the diameter (d). Since the diameter is twice the radius, the radius (r) is equal to d/2.
The volume of the cylinder is given as 11.3 m^3, so we can write:
11.3 = π(d/2)^2h
Simplifying, we have:
11.3 = (πd^2h)/4
Rearranging the equation, we get:
h = (4 * 11.3) / (πd^2)
Step 3: Express the surface area as a function of 'd':
Now, we can express the lateral surface area (A_lateral) as a function of 'd':
A_lateral = 2πrh
= 2π(d/2)(4 * 11.3) / (πd^2)
= 2 *((4 * 11.3) / d)
The total surface area (A_total) is the sum of the area of the two bases and the lateral surface area:
A_total = 2πr^2 + 2πrh
= π(d/2)^2 + 2 *((4 * 11.3) / d)
Step 4: Find the derivative and critical points:
To find the minimum surface area, we differentiate A_total with respect to 'd', and set it equal to zero:
dA_total/d = 0
Deriving A_total:
dA_total/d = 0 + (8 * 11.3) / d^2 - (8 * 11.3) / d^2
Simplifying further:
dA_total/d = 16 * 11.3 /d^2 - 16 * 11.3 /d^2
= 16 * 11.3 /d^2 - 16 * 11.3 /d^2
= 0
Step 5: Solve for 'd':
Simplifying the equation further:
16 * 11.3 = 16 * 11.3
This equation holds true for all values of 'd'.
Hence, there are no critical points, which means that the surface area is minimized for all values of 'd'.
Step 6: Conclusion:
Since the equation holds true for all values of 'd', there is no specific value of 'd' that minimizes the surface area. Therefore, the diameter of the closed cylindrical tank can be any value, and it will have the minimum surface area as long as the volume is 11.3 m^3.