Question

An open rectangular box with volume 2m^3 has a square base. Express the surface area of the box as a function of the length of a side of the base.

Answer 1

the answer the book gave was S(x)= x^2 + (8/x), x>0
but I don't understand this

Answer 2

Start with the base of the box. You know its area is length x width, and that since its a square, the length of the each side of the base are equal to each other. So you can say length x length = the are of the base.

Now each side of the box has the area length x height. Notice that "length" here means the same as the "length" on the base, because they are connected to the base :D.

Think about this, try drawing a picture and make a formula for it. The last bit is just considering how it relates to the 2m^3 volume you've been given, but I think that depends on the level of maths you are in.

Answer 3

calculus 1

Answer 4

but how do you find surface area?

Answer 5

Sorry, working on my own homework. Im not sure I understand your question. The area of a 2d object is its surface area.

Answer 6

Can you express the volume of the box using the figure above?

Answer 7

@mangoes20 Are you still there?

Answer 8

i'm just not understanding at alll

Answer 9

Do you see my figure above?

Answer 10

If you want help with this problem, I'll help you now. If you don't answer, I'm getting off in a few minutes.

Answer 11

ok i'm ready

Answer 12

OK, Look at the figure above.
That is a box with a base and 4 sides but no top.

Answer 13

ok

Answer 14

The surface area of the box is the sum of the areas of all 4 vertical faces and the area of the base.

Answer 15

ok

Answer 16

The base of the box is a square with side length x.
What is the area of the base?

Answer 17

x x y

Answer 18

No. That is the volume of the box which we'll need soon.

The base of the box is a square. The area of a square is the square of the side.
Since the side of the base has length x, the area of the base is x^2.

Answer 19

Now we have the area of the base.

The next thing we need is the area of the 4 vertical sides. Each vertical side is a rectangle with side lengths x and y. What is the area of each vertical side?

Answer 20

xy^2

Answer 21

How do you get xy^2 from multiplying x and y?
Each side is a rectangle x by y, so its area is simply xy.

Answer 22

There are 4 vertical sides, so the total are of vertical sides is 4xy.

Answer 23

oh

Answer 24

Now we have the area of the base is x^2, and the area of all 4 sides is 4xy.
A = x^2 + 4xy

Answer 25

We need the area expressed in terms of x only, but we have y in the expression.
Now we use the volume to get rid of y.

Answer 26

The volume of a box is
V = LWH ( volume = length * width * height)

Answer 27

For this box, we are given the volume is 2 m^3.
The volume is
V = x * x* y
V = x^2y

Answer 28

We are given that the volume is 2 m^2, so we write
x^2y = 2, and we solve for y:
y = 2/(x^2)

Answer 29

Now we go back to our expression for the area, and we replace y with 2/(x^2):

A = x^2 + 4xy
A = x^2 + 4x[2/(x^2)]
A = x^2 + 8x/x^2
A = x^2 + 8/x

Answer 30

wow you actually made that make sense!

Answer 31

That's great. Go over it step by step. It's important to learn the method.

Answer 32

Oh I will

Answer 33

Great.