In a rectangular piece of cardboard with a perimeter of 20ft, three parallel and equally spaces creases are made (so the piece is divided into four equal sections.) The cardboard is then folded to make a rectangular box with open square ends.
a. Write a formula for the volume of the box.
b. Find the domain of a.
.................
. . . . .
. . . . .
. . . . . y
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------4x-------
let the length of the original rectangle be 4x (so that each of the smaller rectangles is x )
let the width be y
After it is folded, the volume would be
y(x^2)
but 8x + 2y = 20
or y = 10 - 4x
so Volume = (10 - 4x)x^2
for b. you ask "what is the domain of a"
there is no a
looks like my diagram did not come out correctly.
It was supposed to be a large rectangle 4x by y, divided into 4 equal rectangles each x by y
(-1,4)(5,8) into general equation form
a. To find a formula for the volume of the box, we first need to understand the dimensions of the box. Let's assume the length of the cardboard is L and the width is W (both in feet).
When the cardboard is folded, the three creases divide it into four equal sections. Since they are equally spaced, the distance between each crease is (W/4) feet.
If we fold the cardboard into a box, the length (L) becomes the height of the box, and the width (W) becomes the length (L) of the box. The distance between the creases (W/4) becomes the width (W) of the box.
Since the box has open square ends, the dimensions of the box are:
Height (H) = L (from the original length of the cardboard)
Length (L) = W (from the original width of the cardboard)
Width (W) = W/4 (distance between the creases)
Now, we can write the formula for the volume (V) of the box:
V = Length x Width x Height
Plugging in the values:
V = W x (W/4) x L
Simplifying the equation, we get:
V = (W^2 x L) / 4
b. To find the domain of the formula, we need to consider the restrictions on the dimensions of the box. The dimensions (W, L) must be positive since we are dealing with lengths. Additionally, the width (W) cannot be zero, as it would result in a box with no volume.
So, the domain for the formula is:
W > 0 and L > 0