Could you please explain this problem step by step, thank you!

You are planning to make an open rectangular box that will hold a volume of 50 cubed feet. What are the dimensions of the box with minimum surface area?

x = length

y = width
h = height
x y h = 50
area bottom = x y
area sides = 2 x h + 2 y h
total area a = x y + 2 x h + 2 y h
so
h = 50/xy

a = x y + 100/y + 100/x
da/dx=x dy/dx+y-100 dy/dx /y^2-100/x^2
for min = 0
dy/dx(x-100/y^2) + y-100/x^2 = 0
x y^2 = 100 and y x^2 = 100
y (100^2/y^4) = 100
y^3 = 100
y = 4.64
x = 100/y^2 = 4.64
h = 2.32

Step 1: Understand the problem

We are tasked with finding the dimensions of an open rectangular box that will have a volume of 50 cubic feet and the minimum surface area.

Step 2: Identify the variables
Let's label the length, width, and height of the box as L, W, and H, respectively. We need to find the values of L, W, and H that will minimize the surface area, given a volume of 50 cubic feet.

Step 3: Relate the variables
The volume of a rectangular box is given by the formula V = L * W * H. In our case, the volume is 50 cubic feet, so we have the equation 50 = L * W * H.

Step 4: Express surface area in terms of the variables
The surface area of an open rectangular box can be divided into five parts: the top, the bottom, and the four sides. Therefore, the surface area (S) is given by the formula S = 2(LW + LH + WH).

Step 5: Reduce variables using the volume equation
To simplify the problem, we need to express one of the variables (L, W, or H) in terms of the other two. We can rearrange the volume equation (50 = L * W * H) to express L in terms of W and H: L = 50 / (W * H).

Step 6: Write the surface area equation in terms of two variables
By substituting L with the expression 50 / (W * H) in the surface area equation S = 2(LW + LH + WH), we get S = 2(50/W + 50/H + WH).

Step 7: Find the derivative
To find the dimensions with minimum surface area, we need to find the critical points of the surface area equation. We can do this by taking the derivative of the surface area equation with respect to one of the variables (W or H).

Step 8: Calculate the derivative
By taking the derivative of the surface area equation S = 2(50/W + 50/H + WH) with respect to H, we get dS/dH = 2(-50/H^2 + W).

Step 9: Set the derivative equal to zero
To find the critical points, we set the derivative equal to zero and solve for H: 0 = 2(-50/H^2 + W), which simplifies to -50/H^2 + W = 0.

Step 10: Solve for H
Rearranging the equation, we get -50/H^2 = -W. Then, dividing both sides by -50/H^2, we get H^2 = 50/W.

Step 11: Solve for W
Taking the square root of both sides of the equation H^2 = 50/W, we get H = sqrt(50/W).

Step 12: Find L using the volume equation
Using the volume equation 50 = L * W * H, we can substitute the values we found for H and W: 50 = L * W * sqrt(50/W). Simplifying the equation, we get 50 = L * sqrt(50W).

Step 13: Solve for L
Dividing both sides of the equation by sqrt(50W), we get L = 50 / sqrt(50W).

Step 14: Determine the dimensions
We have found expressions for L, W, and H in terms of each other. The dimensions of the box with minimum surface area are given by L = 50 / sqrt(50W), W, and H = sqrt(50/W).

So, to find the dimensions of the box with the minimum surface area, you would substitute a value for W in the expressions for L and H, using either algebra or a graphing calculator.

Sure! To solve this problem, we need to find the dimensions of an open rectangular box that has a minimum surface area while holding a volume of 50 cubic feet.

Step 1: Understand the problem
We are trying to find the dimensions of an open rectangular box with minimum surface area. The box needs to have a volume of 50 cubic feet.

Step 2: Identify the variables
Let's assume the length, width, and height of the box as l, w, and h respectively.

Step 3: Formulate the equations
We have two conditions: the volume of the box needs to be 50 cubic feet, and we want to minimize the surface area.

The volume equation is given by:
Volume = length x width x height
50 = lwh

The surface area of an open rectangular box can be obtained by summing the areas of all its sides. Since it is open, we can ignore one of the sides. The surface area equation is given by:
Surface Area = 2lw + lh

Step 4: Solve the problem
To find the dimensions of the box with minimum surface area, we need to minimize the surface area equation while satisfying the volume equation.

We can use the volume equation to express one variable in terms of the other two. Since we want to minimize the surface area, we can differentiate the surface area equation with respect to one of the variables and solve for it.

From the volume equation, we have:
h = 50 / (lw)

Substituting this value of h in the surface area equation, we get:
Surface Area = 2lw + l(50 / lw)
Surface Area = 2lw + 50 / l

To minimize the surface area, we can differentiate the equation with respect to l and set it equal to zero:
d(Surface Area) / dl = 2w - 50 / l^2 = 0

Solving for l, we get:
2w = 50 / l^2
2wl^2 = 50
l^2 = 25 / w

Step 5: Find the dimensions
To find the dimensions, we need to solve for l, w, and h. From the above equation, we have:
l = sqrt(25 / w)

Now, we know that l * w * h = 50.
Substituting the value of l, we have:
sqrt(25 / w) * w * h = 50

Simplifying the equation, we get:
w * sqrt(25 / w) * h = 50
wh = 50

We know that w * h = 50. Therefore, one possible solution is w = 5 and h = 10. Using these dimensions, we can calculate l = sqrt(25/5) = 5.

So, the dimensions of the box with minimum surface area that holds a volume of 50 cubic feet are l = 5 feet, w = 5 feet, and h = 10 feet.

For symmetry reasons, the optimum must have a square base. Let the height of the walls be x and the side length be y

Volume = x^2*y = 50
Area = x^2 +4xy = x^2 + 4x*50/x^2
= x^2 + 200/x
dA/dx = 0 = 2x -200/x^2
x^3 = 100
x = 4.64 ft
y = 2.32 ft