1. a square based prism box has a capacity of 8L. wHAT DIMENSIONS PRODUCE THE MINIMUM SURFACE AREA?

2. what are the dimensions of the square based prism box with maximum volume that can be made using 6oocm^2 of cardboard.
after getting the answer, pls i need you to explain how you got it and the steps you followed. Thank you.

1. To find the dimensions that produce the minimum surface area for a square-based prism box with a capacity of 8L, we need to use the concept of optimization.

Let's assume that the edges of the square base have a length of x, and the height of the prism is h.

The volume of a prism is given by V = Base area * height. In this case, the base area is x^2, and the volume is given as 8L, which is equivalent to 8000 cm^3.

Therefore, V = x^2 * h = 8000.

To find the surface area, we need to consider the five faces of the prism. Two opposite faces have an area of x * h (the lateral faces), and the other four faces have an area of x^2 (the base and top faces). So the surface area (A) is:

A = 2(x * h) + 4(x^2) = 2xh + 4x^2.

Now, we need to express the surface area as a function of a single variable, which allows us to optimize it. In this case, we can express h in terms of x using the volume equation:

h = 8000 / x^2.

Substituting this expression for h into the surface area equation:

A = 2x(8000 / x^2) + 4x^2.

Simplifying further:

A = 16000 / x + 4x^2.

To find the minimum surface area, we can take the derivative of A with respect to x and set it equal to zero:

dA/dx = 0 - 16000 / x^2 + 8x = 0.

Solving this equation gives us:

16000 / x^2 = 8x.

Simplifying further:

2000 = x^3.

Taking the cube root of both sides:

x = ∛(2000) ≈ 12.59 cm.

Now that we have the value of x, we can substitute it back into the volume equation to find the corresponding value of h:

h = 8000 / x^2 = 8000 / (12.59)^2 ≈ 50.17 cm.

Therefore, the dimensions of the square-based prism box that produce the minimum surface area for a capacity of 8L are approximately 12.59 cm (length of each side of the square base) and 50.17 cm (height).

2. To find the dimensions of the square-based prism box with the maximum volume that can be made using 600 cm^2 of cardboard, we need to use the same concept of optimization.

Let's assume that the edges of the square base have a length of x, and the height of the prism is h.

The surface area of the prism is given by A = 2(x * h) + 4(x^2). In this case, the surface area is given as 600 cm^2.

Therefore, 2(x * h) + 4(x^2) = 600.

We also know that the volume of the prism is V = x^2 * h.

To solve for h, we can express h in terms of x from the surface area equation:

2(x * h) + 4(x^2) = 600.
2(x * h) = 600 - 4(x^2).
h = (600 - 4(x^2)) / 2x.

Substituting this expression for h into the volume equation:

V = x^2 * [(600 - 4(x^2)) / 2x].
V = (600x - 4x^3) / 2.
V = 300x - 2x^3.

To find the maximum volume, we take the derivative of V with respect to x and set it equal to zero:

dV/dx = 300 - 6x^2 = 0.

Solving this equation gives us:

6x^2 = 300.
x^2 = 50.
x = √(50) ≈ 7.07 cm.

Now that we have the value of x, we can substitute it back into the expression for h:

h = (600 - 4(x^2)) / 2x.
h = (600 - 4(7.07)^2) / 2(7.07).
h ≈ 28.29 cm.

Therefore, the dimensions of the square-based prism box with the maximum volume that can be made using 600 cm^2 of cardboard are approximately 7.07 cm (length of each side of the square base) and 28.29 cm (height).