A cardboard box without a top is to have volume 500000 cubic cm. Find the dimensions which minimize the amount of material used. List them in ascending order.

To find the dimensions that minimize the amount of material used for a cardboard box without a top, we need to determine its surface area and then optimize it.

Let's assume the dimensions of the box are length (L), width (W), and height (H). Since the box has no top, it will have only five sides: a bottom, two sides, a front, and a back.

The surface area of the box can be calculated using the following formula:

Surface Area = 2(LW) + LH + WH

We need to minimize this surface area while keeping the volume of the box constant at 500,000 cubic cm. The volume formula for a box is:

Volume = LWH

Since we know the volume (500,000 cubic cm), we can express one of the variables (L, W, or H) in terms of the other two.

Let's solve the volume equation for L:

L = 500,000 / (WH)

Now, substitute this value of L in the surface area formula:

Surface Area = 2[(500,000 / (WH))W] + (500,000 / (WH))H + WH

Simplifying further:

Surface Area = (1,000,000 / H) + (1,000,000 / W) + WH

To minimize the surface area, we need to differentiate it with respect to both H and W, and then equate the derivatives to zero:

∂Surface Area / ∂H = -1,000,000 / H^2 + W = 0 -- (1)
∂Surface Area / ∂W = -1,000,000 / W^2 + H = 0 -- (2)

Solving equations (1) and (2) simultaneously will give us the optimal values of H and W.

From equation (1), we get:

W = 1,000,000 / H^2

Substituting this value of W in equation (2), we have:

-1,000,000 / (1,000,000 / H^2)^2 + H = 0

Simplifying further:

-1,000,000 / (1,000,000^2 / H^4) + H = 0
-1,000,000H^4 / 1,000,000^2 + H = 0
-H^4 + 1,000,000^2H = 0

Factoring out H:

H(H^3 - 1,000,000^2) = 0

This equation has two possible solutions:

Solution 1: H = 0

This is not a valid solution since the dimensions cannot be zero.

Solution 2: H^3 - 1,000,000^2 = 0

Taking the cube root of both sides:

H = 1000

Substituting this value of H back into equation (1):

W = 1,000,000 / H^2
W = 1,000,000 / (1000)^2
W = 1,000 / 1
W = 1000

Now, substitute the values of H and W into the volume equation to find L:

Volume = LWH
500,000 = L(1000)(1000)
500,000 = L x 1,000,000
L = 500,000 / 1,000,000
L = 0.5

So, the dimensions of the box that minimize the amount of material used are:

Length(L): 0.5 cm
Width(W): 1000 cm
Height(H): 1000 cm

Listing them in ascending order:

Dimensions: 0.5 cm x 1000 cm x 1000 cm

To minimize the amount of material used for a cardboard box without a top, we can consider different dimensions of the box and calculate the amount of material required for each case. The dimensions that minimize the amount of material used will be the ones that result in the smallest surface area of the box.

Let's assume the length, width, and height of the box are L, W, and H, respectively. We know that the volume of the box is 500000 cubic cm, so we have the equation:

L * W * H = 500000

We also know that the box does not have a top, so the surface area of the box (including the bottom) is:

2(LW + LH) = 2LW + 2LH

To minimize the amount of material used, we need to minimize this surface area.

To find the dimensions, we can use calculus. Let's solve for one of the variables, for example, H, in terms of the other two variables:

H = 500000 / (L * W)

Substituting this value of H in the surface area equation, we get:

2LW + 2L(500000 / LW) = 2LW + 1000000/L

To find the minimum of this function, we can take the derivative with respect to L and set it equal to zero:

dA/dL = 2W - 1000000/L^2 = 0

Solving for L, we have:

2W = 1000000/L^2

L^2 = 1000000/(2W)

L = √(500000/W)

Similarly, taking the derivative with respect to W and setting it equal to zero, we get:

2L = 1000000/W^2

W^2 = 1000000/2L

W = √(500000/L)

Now we have expressions for L and W in terms of each other. To find the minimum, we can substitute these expressions into the volume equation:

L * W * H = 500000

(√(500000/W)) * W * (√(500000/W)) = 500000

Simplifying,

(500000/W) * W = 500000

This gives us W = √500000 = 500√2

Substituting this value of W into the expression for L,

L = √(500000/W) = √(500000/(500√2)) = √(1000/√2) = 10√2

Finally, to find the missing dimension, H, we can substitute the values of L and W into the equation:

H = 500000 / (L * W) = 500000 / (10√2 * 500√2) = 1/10

Therefore, the dimensions of the box that minimize the amount of material used, listed in ascending order, are:

1/10 cm, 10√2 cm, 500√2 cm