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(Solved): A box with a square base and open top must have a volume of 219,488cm^(3). We wish to find the dimen ...



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))=

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