A rectangular tank with a square base, an open top, and a volume of 4,000 ft cubed is to be constructed of sheet steel. Find the dimensions of the tank that has the minimum surface area.
I'm not understanding how to get started and find the optimization function (for any of these problems).
To determine the dimensions of the tank that minimize the surface area, we need to first set up an optimization problem. In this case, we want to minimize the surface area of the tank subject to the constraint that its volume is 4,000 ft³.
Let's start by defining the variables in the problem:
- Let x represent the side length of the square base of the tank.
- Let h represent the height of the tank.
Now, let's set up the equations:
1. Volume equation: The volume of a rectangular tank is given by the formula V = base area × height. Since the base is square, the base area is x². Therefore, the volume equation is x²h = 4000.
2. Surface area equation: The surface area of the tank consists of the area of the base (x²) and four rectangular sides (2hx). Therefore, the surface area equation is A = x² + 4hx.
We want to minimize the surface area, which means we need to express the surface area A as a function of a single variable. We can use the volume equation to express h in terms of x and substitute it into the surface area equation:
x²h = 4000 (volume equation)
h = 4000/x²
A = x² + 4hx (surface area equation)
A = x² + 4x(4000/x²) (substituting h)
Simplifying the equation:
A = x² + 16000/x
Now we have the surface area A expressed as a function of x. To find the values of x that minimize A, we can take the derivative of A with respect to x and set it equal to zero:
dA/dx = 2x - 16000/x² (derivative of A with respect to x)
0 = 2x - 16000/x²
To solve this equation, we can multiply both sides by x² to eliminate the denominator:
0 = 2x³ - 16000
Now, we have a cubic equation that we can solve for x. By finding the real positive roots, we can then plug them back into the volume equation to find the corresponding values of h.
Once we have the values of x and h, we can calculate the minimum surface area by substituting those values into the surface area equation A = x² + 4hx.
To find the dimensions of the tank that minimize the surface area, we need to identify an optimization function and apply the appropriate techniques to solve it.
Let's denote the side length of the square base as x and the height of the tank as h. The volume V of the tank is given as 4,000 ft³.
Step 1: Identify the optimization function
We know that the surface area A of the rectangular tank is given by the sum of the areas of the base and the four sides:
A = x² + 4xh
We need to minimize A subject to the constraint that the volume V is equal to 4,000 ft³.
Step 2: Express the constraint in terms of the variables
The volume of the rectangular tank is given by:
V = x²h
Since V = 4,000 ft³, we can express h in terms of x:
h = V / x²
h = 4000 / x²
Step 3: Substitute h into the optimization function
Substituting h into A:
A = x² + 4x(4000 / x²)
A = x² + 16000 / x
Step 4: Differentiate the optimization function
To find the minimum of A, we need to differentiate A with respect to x and set the derivative equal to zero:
dA/dx = 2x - 16000 / x² = 0
Step 5: Solve for x
To solve this equation for x, multiply both sides by x²:
2x³ - 16000 = 0
2x³ = 16000
x³ = 8000
x = ∛(8000)
x = 20 ft
Step 6: Find the corresponding value of h
We can substitute x = 20 ft into the expression for h:
h = 4000 / (20²)
h = 10 ft
Therefore, the dimensions of the tank with the minimum surface area are 20 ft by 20 ft by 10 ft.
side = x
height = y
volume = x^2y = 4000 so y = 4000/x^2
area = A = x^2 + 4 x y
A = x^2 + 4 x (4000/x^2)
A = x^2 + 16000/x
min or max when dA/dx = 0
0 = 2x -16000/x^2
16000 = 2 x^3
x^3 = 16000
x = 25.2
y = 4000/x^2 = 6.3