A closed cardboard box is designed to hold a volume of 288 cm^3 . The length is 3 cm and the width x cm .Show that the total surface area A cm^2 is given by A=6x^2+768/x and find the dimensions of the box which will make A a minimum. [8 Marks]
L * x * h = 288 so hx = 96
A = 2*L*x + 2*h * L + 2 *h*x = 2 (Lx + Lh + hx) = 6 x + 6 h + 192
A = 6 x + 576/x + 192
hmm
dA/dx = 6 - 576/x
= 0 for min or max
x = 576/6 = 96
h = 1
L = 3
well 96 * 1 * 3 = 288 so that checks
w = x
l = 3
let the height be h
wlh = V
3xh = 288
xh = 96
h = 96/x
Area = 2(wl + lh + wh)
= 2(3x + 3(96/x) + x(96/x) )
= 6x + 576/x + 192
or (6x^2 + 576 + 192x)/x <==== which does not match what is given in the question
dA/dx = 6 - 576/x^2 = 0 for a minimum of A
576/x^2 = 6
6x^2 = 576
x^2 = 96
x = √96 = 4√6 or appr 9.8 cm
To find the total surface area A of the closed cardboard box, we need to determine the area of each face and sum them up.
The box has six faces: a top face, a bottom face, a front face, a back face, a left face, and a right face.
1. Top and bottom face: The area of each top and bottom face is equal to the length multiplied by the width. Since the length is given as 3 cm and the width is x cm, the area of each top and bottom face is 3x cm^2.
2. Front and back face: The area of each front and back face is equal to the length multiplied by the height. However, the height is not given, so we need to express it in terms of the given width, x. The volume of the box is given as 288 cm^3, and the volume of a rectangular prism is calculated by multiplying the length, width, and height. Therefore, we have the equation:
3 cm * x cm * height = 288 cm^3
Simplifying, we obtain:
x * height = 96 cm^3
height = (96 cm^3) / x
Applying this to the front and back face, each face has an area of 3 * (96 cm^3) / x cm^2, which simplifies to 288/x cm^2.
3. Left and right face: The area of each left and right face is equal to the width multiplied by the height, which is given as (96 cm^3) / x. Therefore, each face has an area of x * (96 cm^3) / x cm^2, which simplifies to 96 cm^2.
Now, we can sum up the areas of all six faces to get the total surface area A:
A = 2 * (3x) + 2 * (288/x) + 2 * 96 cm^2
A = 6x + 576/x + 192 cm^2
A = 6x + 576/x + 192 cm^2
Thus, we have the equation for the total surface area A: A = 6x^2 + 576 + 192/x cm^2.
To find the dimensions of the box that minimize A, we need to find the value of x that minimizes the given equation.
To do this, we can differentiate the equation with respect to x and set it equal to zero:
dA/dx = 12x - 192/x^2 = 0
Multiplying through by x^2 gives:
12x^3 - 192 = 0
12x^3 = 192
x^3 = 16
x = ∛16
x = 2 cm
Therefore, the width of the box that minimizes the total surface area A is x = 2 cm.