If an open box has a square base and a volume of 111 in.3 and is constructed from a tin sheet, find the dimensions of the box, assuming a minimum amount of material is used in its construction. (Round your answers to two decimal places.)

  6.06 inches square by 3.03 inches high

Step-by-step explanation:

An open-top box will have a minimum surface area when its lateral area is double its base area. That is, left-area + right-area = front-area + back area = bottom area.

With that in mind, the simplest way to compute the dimensions is to find the base edge length as the cube root of double the volume.

  b = ∛(2×111 in³) ≈ 6.06 in

  h = b/2 = 3.03 in . . . . . there are 4 faces; each is half the area of the base

The box is 6.06 inches square and 3.03 inches high.

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Solution using derivatives

Let b represent the edge length of the base. The volume is ...

  V = b^2h

Then the height is ...

  h = V/b^2

  h = 111/b^2

and the total surface area is ...

  A = b^2 + 4bh = b^2 +4·111/b

The area is maximized when the derivative is zero:

  dA/db = 0 = 2b -444/b^2

Solving for b, we get ...

  2b^3 = 444

  b^3 = 222

  b = ∛222 ≈ 6.06 . . . . . . as above

when rotating a figure 180 degrees, imagine that you are able to take the figure and turn it clockwise or counterclockwise around a center point. to rotate a figure 180 degrees, you will need to apply the rule (x, y) → (-x, -y). start by using a coordinate grid with coordinates for each vertex of the figure. the center point of the coordinate grid is located at (0, 0), which is what you will rotate the figure around. write down the original coordinates of the shape you are going to rotate. then, apply the rule. for example, coordinate (1, 2) becomes (-1, -2). another example would be, coordinate (-4, -2) becoming (4, 2). after you change each original coordinate to the rotated coordinate, you will draw your new figure.

step-by-step explanation:

second option

step-by-step explanation:

the measures of a triangle equal 180. use that to find x.

2x+3x+4x=180

9x=180

9x/9=180/9

x=20

now you know x, you know angle a is 40°, angle b is 60°, angle c is 80°.

because angle a is 40°, the first option doesn't work.

because angle b is 60°, the second option does work.

when you add angle a and b, you get 100°. complementary means it equals 90°, so the third option doesn't work.

when you add angle a and c, you get 120°, which is not 100.