A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in. by 20 in. by cutting out equal squares of side x at each corner and then folding up the sides as in the figure. Express the volume of the box as a function of x.
Solution:
From the diagram, we know that x is also the height of the box. Then, the length,
L=20-2xand the width,
W=12-2x.
The volume of the box,
V=LWx,
meaning that
V=(20-2x)(12-2x)(x).
Expanding the brackets and simplifying leads us to,
V=4x^{3}-64x^{2}+240x
To figure out the domain, the following conditions must be true. The length,
L>0\Leftrightarrow 20-2x>0\Leftrightarrow x<10and the width,
W>0\Leftrightarrow 12-2x>0\Leftrightarrow x<6, x>0.
Combining these leads us to, 0<x<6
thank you very much for posting these solutions. it really helped gain a deeper understanding of the problem.
Do you always need to find domain with volume problems?
Only if the question asks for it.
Well done
here x is unknown how we will fiind the x please
You would need more data.
What number did use to calculate to get 64x
I am not exactly sure of your question, you are just expanding the brackets. Multiply these out: (20−2x)(12−2x)(x).