Question 202658
a)



{{{V(s)=s^3-13s^2+54s-72}}} Start with the given function.



{{{V(10)=(10)^3-13(10)^2+54(10)-72}}} Plug in {{{s=10}}} (ie replace each "s" with 10).



{{{V(10)=1000-13(10)^2+54(10)-72}}} Cube {{{10}}} to get {{{1000}}}.



{{{V(10)=1000-13(100)+54(10)-72}}} Square {{{10}}} to get {{{100}}}.



{{{V(10)=1000-1300+54(10)-72}}} Multiply {{{-13}}} and {{{100}}} to get {{{-1300}}}.



{{{V(10)=1000-1300+540-72}}} Multiply {{{54}}} and {{{10}}} to get {{{540}}}.



{{{V(10)=168}}} Combine like terms.



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b)



{{{V=LWH}}} Start with the volume equation.



{{{s^3-13s^2+54s-72=L(s-6)H}}} Plug in the given expressions



{{{(s^3-13s^2+54s-72)/(s-6)=LH}}} Divide both sides by {{{s-6}}}.



{{{((s-6)(s^2-7s+12))/(s-6)=LH}}} Factor the numerator



{{{(cross((s-6))(s^2-7s+12))/cross((s-6))=LH}}} Cancel out the common terms.



{{{s^2-7s+12=LH}}} Simplify



{{{LH=s^2-7s+12}}} Rearrange the equation



{{{LH=(s-3)(s-4)}}} Factor



So the new length and height are {{{s-3}}} and {{{s-4}}}



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c)


I'll let you attempt this one on your own. Repost if you still need help.



Note: the answer is going to be the same as the answer in part a) (so you'll have something to check)