express total surface area of a box in terms of the height of the box?

Question

A box has rectangular top, bottom, and sides. The top and bottom are square. The volume must be 4 cubic meters. Express the total surface area A of the box in terms of the height h (in meters) of the box.
thanks for your help!

Answer 1

Awesome question

Volume = Height * base * depth

since the top and bottom are square, base and depth are equal. We will call them B

H * B * B = 4
divide both sides by h

b^2 = 4h
b = 2*sqr(h)

Top and bottom each = (2*(sqr(h))^2 = 4h
Each of the 4 sides = h * 2(sqr(h)) = 2h*(sqr(h))
Total area = 2*4h +4*(2h*(sqr(h))
area = 8h + 8h(sqr(h))
area = 8h(1+sqr(h))

Answer 2

Let x = length of one side of the top or bottom square
Let h = height of the box

Volume = Area of top or bottom square * height
= x*x*h = hx^2 (h times x squared)

Volume = 4
Hence,

hx^2 = 4 (1)

A = 2*(Area of top or bottom square) + 4*(area of one side of the box)

(The top and bottom have the same area so u just multiply the top area by 2 and the sides of the box all have the same area so u just multiply the area of one of them by 4)

Area of top square = x^2
Area of one side = xh

Substituting them into the area formula:

A = 2x^2 +4xh (2)

Since we want the area in terms of h we have to to find x in terms of h and then sub it into equation (2) above. We can find x in terms of h by transposing equation (1)

hx^2 = 4 (1)
x^2 = 4/h
x = 2/squareroot(h) (taking the square root of both sides)
= 2h^(-1/2) (I like to write it in index form)

Substituting x into equation (2)

A = 2x^2 + 4xh
A = 2*(4/h) +4*h*2*h(-1/2)

A = 8/h + 8h^(1/2)

A = 8/h + 8squareroot(h)

Answer 3

Let
w = width
L = length
h = height
V = volume
S = surface area

Given
V = 4
length = width

Find S in terms of h.

We have

L = w

V = Lwh = w²h
w² = V/h
w = √(V/h)

S = 2Lw + 2Lh + 2wh = 2w² + 2wh + 2wh = 2w² + 4wh
S = 2(V/h) + 4h√(V/h) = 2V/h + 4√(Vh)
S = 2*4/h + 4√(4h) = 8/h + 8√h

Answer 4

let x be the side of square top and bottom;
let h be the height
volume =(X)(X)(h) =4

h = 4 / square of x

total sutface area is 2(square of X) + 4 (X) (h)

by substitution;

=16 / h