Find the volume of a box as a function of the width of base of the box?
If 12,000 cm^2 of material is available to make a box with a square base and an open top, find the volume of the box as a function of the width if the base of the box
Find the volume of a box as a function of the width of base of the box?
If 12,000 cm^2 of material is available to make a box with a square base and an open top, find the volume of the box as a function of the width if the base of the box
did you draw a diagram? (see the attached).
let the width of the base be $\displaystyle x$ and the height of the box be $\displaystyle y$
thus the volume (length times width times height) is given by
$\displaystyle V = x^2 y$
Now we know the surface area is 12000, this is the sum of the areas of all the faces, so
$\displaystyle 12000 = \underbrace{x^2}_{\text{area of base}} + \underbrace{4xy}_{\text{area of the 4 sides}}$
that is $\displaystyle 12000 = x^2 + 4xy$
now can you answer the problem?
Hello, realintegerz!
If 12,000 cm² of material is available to make a box
with a square base and an open top, find the volume of the box
as a function of the width of the base of the box.Code:*-------* /| /| / | / | *-------* | | | | | | | h| | * | | / | |/ x *-------* x
Let $\displaystyle x$ = side of the square base.
Let $\displaystyle h$ = height of the box.
The panels comprising the box is made up of:
. . a base with area $\displaystyle x^2$
. . and four sides with area $\displaystyle xh$ each.
The total surface area is: .$\displaystyle x^2 + 4xh$
We are told that the total surface area will be 12,000 cm².
So we have: .$\displaystyle x^2 + 4xh \:=\:12,\!000 \quad\Rightarrow\quad h \:=\:\frac{12,\!000-x^2}{4x}$ .[1]
The volume of the box is: .$\displaystyle V \;=\;x^2h$
Substitute [1]: .$\displaystyle V \;=\;x^2\left(\frac{12,\!000-x^2}{4x}\right) \quad\Rightarrow\quad \boxed{V \;=\;\frac{1}{4}(12,\!000x - x^3)}$