A box with a square base and open top must have a volume of
219,488cm^(3)
. We wish to find the dimensions of the box that minimize the amount of material used. First, find a formula for the surface area of the box in terms of only
x
, the length of one side of the square base. [Hint: uco the volume formula to express the height of the box in terms of
x
.]
A(x)=
Next, find the derivative,
A^(')(x)
.
A^(')(x)=
Part 2 of 2 At this point we know that
A^(')(x)=2x-4(219488)/(x^(2))
. Now, find the value of
r
that makes the derivative zero.
A^(')(x)=0
when
x=
We next have to make sure that this value of
x
gives a minimum value for the surface area. Let's use the second derivative test. Find
A^('')(x)
.
A^('')(x)=
Evaluate
A^('')(x)
at the
x
-value you gave above. NOTE: Your last answer above should be positive; this means that the graph of
A(x)
is concave up around that value, so the zero of
A^(')(x)
indicates a local minimum for
A(x)
. Question Help: Written Example Let
log(A)=-1,log(B)=2
, and
log(C)=8
. Evaluate the following logarithms using logarithmic properties.
log((A^(5))/(\sqrt(B)))=
◻
log((B)/(C^(4)))=
◻
log((A)/(B^(2)C))=
◻