An open box is made from a rectangular piece of material 12" by 16" by cutting out equal squares from each....

Question

corner and turing up the sides. Let x be the length of each side of the squares from each corner. Write the volume V of the box as a function of x....
Then find the surface area
any help would be great!!

Answer 1

For volume of a rectangular prism, we have base * width * height. If you draw the problem, you'll see that the height is x, the width is 12-2x and the base is 16-2x. So we get a volume:

V = x (12-2x)(16-2x) = x(192 - 56x + 4x^2) = 4x^3 - 56x^2 + 192x

For surface area, we need the sum of the areas of the 5 sides (it has no top).

s1 = s3 = x(12-2x) = 12x-2x^2
s2 = s4 = x(16-2x) = 16x - 2x^2
s5 = (12-2x)(16-2x) = 4x^2 - 56x + 192

so total surface area would be:

2(12x-2x^2) + 2(16x-2x^2) + (4x^2 - 56x + 192) = 192 - 4x2

(we could have figured surface area much more easily since we know the surface area is 12x16 with 4 squares cut from it, or 192 - 4(x*x) = 192-4x^2)

Hope this helps! :)

Answer 2

V = x(16 - 2x)(12 - 2x)
= 4x^3 - 56x^2 + 192x cubic inches

SA = (12 - 2x)(16 - 2x) + 2(12 - 2x)x + 2(16 - 2x)x
= 4x^2 - 56x + 192 + 24x -4x^2 + 32x - 4x^2

Group like terms to find
SA = (192 - 4x^2) square inches

Answer 3

(12-2x)(16-2x)x is the volume
or
(4x^2-56X+192)x
or
4x^3 - 56x^2 + 192x

For the surface area it would be
(12-2x)(16-2x) + 2(x)(12-2x) + 2(x)(16-2x)