Acompany that manufactures gift boxes wants to know how many different-sized boxes having a volume of 50 cubic centimeters it can make if the dimensions must be whole centimeters.
a. list all the possible whole number dimensions for the box.
b. which possibility requires the least amount of material to make?
c. which box would you recommend the company use? why?

a) Box1: 1 cm × 1 cm × 50 cm

Box2: 1 cm × 25 cm × 2 cm

Box3: 1 cm × 5 cm × 10 cm

Box4: 5 cm × 5 cm × 2 cm

b) Box 4 needs lees material. S=90cm²

c) I would recommend to use the 5 cm×5 cm×2cm box because it uses less material.

Step-by-step explanation:

a) The number 50 can be represented as:

50=1×1×50

50=1×5×10

50=1×25×2

50=5×5×2

Hence, these are all the possibilities of the dimensions of the boxes:

Box1: 1 cm × 1 cm × 50 cm

Box2: 1 cm × 25 cm × 2 cm

Box3: 1 cm × 5 cm × 10 cm

Box4: 5 cm × 5 cm × 2cm

b)

answer is:

1.75 pounds

for the friend request.

when a figure, such as triangle (xyz) is reflected over the line y = x to create x'y'z' the x-coordinates and y-coordinates of the original image (pre-image) will be reversed for the image. (e.g., (x, y) > (y, x)) this means that the pre-image position of the x- and y-coordinates switch places which in turn provides a mirror like image of the original xyz.  

i notice that when i draw a line segment from x to x', that the points are parallel from each other -- they are exactly across from each other and do not intersect with any other points.  

yes, think i would see the same characteristic if you drew the line segment connecting y with the reflecting line and then y' with the reflecting line because of the essence of reflections. because it is a mirror image, each point was effected the same way which leads me to conclude that they same would apply to y and y'.