A square sheet of cardboard 100 cm by 100 cm is to have corners of side length x cut out. These cutouts will create flaps that can be folded up to form a box with no lid. The length and width of the box are given by the expression (100 – 2x), and the height of the box is x, where x is in centimeters.) Find the volume and surface area of the box for each x-value.

Question

A square sheet of cardboard 100 cm by 100 cm is to have corners of side length x cut out. These cutouts will create flaps that can be folded up to form a box with no lid. The length and width of the box are given by the expression (100 – 2x), and the height of the box is x, where x is in centimeters.) Find the volume and surface area of the box for each x-value.

Answer 1

Did you make a sketch?

Then clearly you can see that

Volume = x(100-2x)^2
SA = (100 - 2x)^2 + 4(x)(100-2x)

"for each x-value", you gave none, but if you want only whole numbers
and obviously 100-2x > 0, or 0 < x < 50
here is one example:
let x = 10
volume = 10(100-20)^2 = 64,000 cm^3
SA = (100-20)^2 + 4(10)(80) = 9,600 cm^2

Answer 2

V = L W H

The length and width are equal so:

V = L ∙ W ∙ H = ( 100 - 2 x ) ∙ ( 100 - 2 x ) ∙ x = x ( 100 - 2 x )²

Sufrace area:

SA = base + 4 sides = ( 100 - 2 x )² + 4 ∙ ( 100 - 2 x ) ∙ x =

( 100 - 2 x ) ∙ [ ( 100 - 2 x ) + 4 x ] = ( 100 - 2 x ) ∙ ( 100 + 2 x )

Since:

(a + b ) ( a - b ) = a² - b²

SA = ( 100 - 2 x ) ∙ ( 100 + 2 x ) = 100² - ( 2 x )²

Answer 3

To find the volume and surface area of the box for each x-value, we need to determine the dimensions of the box first.

Given that the length and width of the box are given by the expression (100 – 2x) and the height of the box is x, we can set up the following equations:

Length of the box (L) = 100 - 2x
Width of the box (W) = 100 - 2x
Height of the box (H) = x

1. Volume of the box:
The volume of a rectangular box is given by the formula V = L * W * H. Substituting the given expressions, we have:

V = (100 - 2x) * (100 - 2x) * x

Simplifying this expression will give us the volume of the box in terms of x.

2. Surface area of the box:
The surface area of a rectangular box can be calculated by adding the areas of its six sides. Here, we have four sides with dimensions L * H, one side with dimensions W * H, and one side with dimensions L * W. So, the surface area (A) can be expressed as:

A = 2(L * H) + 2(L * W) + 2(W * H)

Substituting the given expressions, we have:

A = 2(100 - 2x * x) + 2(100 - 2x * x) + 2(100 - 2x * x)

Again, simplifying this expression will give us the surface area of the box in terms of x.

Now, if you provide a specific value for x, I can help you calculate the volume and surface area of the box corresponding to that x-value.