Pre-Calc help?

Question

In a rectangular piece of cardboard with perimeter 20 ft. Three parallel and equally spaced creases are made. The cardboard is then folded to make a rectangular box with open-square ends.
a. Show that the volume of the box is V(x) = x^2(10-4x)
b. What is the domain of V?
c. Find the approximate value of x that maximizes the volume. Then give the approximate maximum volume.

Answer 1

Part a)
base = x by x
top = x by x
height = h
When opened out to form a rectangle :-
top of rectangle = 4x
bottom of rectangle = 4x
height of rectangle = h = (20 - 4x) / 2 = 10 - 4x
V = x ² (10 - 4x)

Part b)
Domain is 0 < x < 2.5

Part c)
V(x) = 10 x ² - 4 x ³
V `(x) = 20 x - 12 x ² = 0 for max. value
12 x² - 20 x = 0
4x (3x - 5) = 0
x = 5 / 3 is required answer
1ft 8 in side maximises the volume.

Answer 2

a) If the cardboard dimensions are L and W, then 2L+2W = 20, so L+W=10. Three equally-spaced creases would split the width up into 4 sections. If the distance in feet between each crease is x feet, then the width is 4x, the length is 10 - 4x, and the folded box has dimensions x, x, and , 10 - 4x. This gives the volume of V(x) = x^2 (10 - 4x).

b) The volume can't be 0 or less so,
x^2 (10 - 4x) > 0
10 - 4x > 0
10 > 4x
x < 10/4
x < 5/2
It also makes no sense if x < 0, so our range is 0 < x < 5/2.

c) Take the derivative of V(x) with respect to x, set it equal to zero, and solve for x. Then use that to find the associated volume.

Answer 3

I can at least help with a.
It might help if you take a sheet of paper and fold it as it says so you can visualize it. Because the creases are equally spaced, you can have each smaller piece be represented by x. For the length, you have 4x (the 4 smaller pieces add up to 4x). For the width you have y. Therefore, the perimeter is 4x + y + 4x + y=20. Simplify this to 8x + 2y =20
For the volume, it will be (length)(width)(height). Once again, if you have it folded, you can see it much easier. Length is y, and width and height are each x. So, Volume = (x)(x)(y) use the perimeter equation to solve for y, so 2y=20-8x, and y=10-4x. Plug in y and you get Volume = (x)(x)(10-4x). Simplify and you get Volume = X^2(10-4x)