well, you know l = w because the base is square.
v = w^2 h = 300 mL = .3 *10^-3 m^3
so h = 3 * 10^-4 /w^2
cardboard area
bottom and top = 2 w^2
sides = 4 w h
so
A = 2 w^2 + 4 w h
A = 2 w^2 + 4 w (3*10^-4)/w^2
A = 2 w^2 + 12*10^-4 /w
dA/dw = 4 w -1.2*10^-3/w^2
= 0 fo min
4 w^3 = 1.2 * 10^-3
w^3 = .3 * 10^-3
w = .67 * 10^-1 meters
= 6.7 centimeters
h = 3*10^-4/w^2
h = 3*10^-4 / .45 *10^-2 = 6.7 *10^-2
= 6.7 centimeters
In other words, the answer is a cube.
This is the question:
"A new fruit juice (like a Popper) is to be marketed in a new container in the shape of a rectangular prism. The cardboard container is to have a square base and is to contain 300 mL of juice. What should be the dimensions of the container be if the amount of cardboard used in its construction is to be a minimum? Disregard waste and overlap."
All I have so far is:
Volume = l x w x h
Therefore:
300 = l x w x h
I honestly have no idea where to go from here.
Do I eventually have to derive an equation and then let that = 0 and then factorise?
I would really appreciate some help, guys.
Thank you in advance!
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