a sheet of cardboard is to be folded up to make a closed box. If the cardboard measures 55 cm by 80cm calculate the maximum volume of the box that can be made from this sheet of cardboard
let each side of the squares to be cut out be x cm
length = 80-2x
width = 55-2x , 0 < x < 27.5
height = x
V = x(80-2x)(55-2x)
= 4x^3 -270x^2 + 4400x
dV/dx = 12x^2 - 540x + 4400
= 0 for a max of V
3x^2 - 135x + 1100 = 0
x = (135 ± √5025)/6
take it from there (I get x = appr 10.69)
To determine the maximum volume of a box that can be made from a sheet of cardboard, we need to consider the dimensions of the cardboard and the geometry of the box.
First, let's understand the dimensions of the cardboard sheet. We are given that the cardboard measures 55 cm by 80 cm.
Next, we have to determine how to fold the sheet to create a closed box. Generally, a box is formed by folding up the sides of the sheet and connecting them. In this case, let's assume that the box is formed by folding the longer side (80 cm) to create the base, and folding the shorter side (55 cm) to create the height of the box.
To calculate the maximum volume, we need to find the dimensions that maximize the volume. Let's call the dimensions of the base of the box length (L), width (W), and height (H).
Since the longer side of the cardboard (80 cm) is used as the base, we have L = 80 cm. And since the shorter side of the cardboard (55 cm) is used as the height, we have H = 55 cm.
Now, we need to determine the width, W, of the base to maximize the volume. To find the maximum volume, we need to find the dimensions that maximize the product of length, width, and height.
So, the volume (V) of the box can be calculated as: V = L * W * H.
Now, since L = 80 cm and H = 55 cm, the volume can be expressed as: V = 80 cm * W * 55 cm.
To maximize the volume, we need to find the value of W that maximizes the product 80 * W * 55.
The maximum value of a product occurs when the two factors being multiplied are equal. In this case, since 80 and 55 are fixed values, the maximum value of the product occurs when W is also equal to 80 cm, i.e., W = 80 cm.
Therefore, the maximum volume (V_max) of the box that can be made from the given cardboard sheet is given by: V_max = 80 cm * 80 cm * 55 cm.
Calculating this, we have: V_max = 352,000 cm^3.
Hence, the maximum volume of the box that can be made from this sheet of cardboard is 352,000 cm^3.