Question 149518

Let {{{L[c]}}}, {{{W[c]}}}, and {{{H[c]}}} be the length width and height of Charlie's box.


So the volume of Charlie's box is

{{{V=L[c]*W[c]*H[c]}}}


Also, the surface area of Charlie's box is


{{{SA=2*L[c]*W[c]+2*L[c]*H[c]+2*W[c]*H[c]}}}



{{{SA=2(L[c]*W[c]+L[c]*H[c]+W[c]*H[c])}}} Factor out the GCF 2.


--------------------


Now let {{{L[l]}}}, {{{W[l]}}}, and {{{H[l]}}} be the length width and height of Lucy's box.



Since Lucy's box has " dimensions twice as large as Charlie's", this means that {{{L[l]=2L[c]}}}, {{{W[l]=2W[c]}}}, and {{{H[l]=2H[c]}}}




So the volume of Lucy's box is

{{{V=L[l]*W[l]*H[l]}}}



{{{V=2L[c]*2W[c]*2H[c]}}} Now plug in {{{L[l]=2L[c]}}}, {{{W[l]=2W[c]}}}, and {{{H[l]=2H[c]}}}



{{{V=8L[c]*W[c]*H[c]}}} Multiply



So the volume of Lucy's box is 8 times larger than Charlie's box.





Also, the surface area of Lucy's box is


{{{SA=2*L[l]*W[l]+2*L[l]*H[l]+2*W[l]*H[l]}}}



{{{SA=2(L[l]*W[l]+L[l]*H[l]+W[l]*H[l])}}} Factor out the GCF 2.


{{{SA=2(2L[c]*2W[c]+2L[c]*2H[c]+2W[c]*2H[c])}}} Now plug in {{{L[l]=2L[c]}}}, {{{W[l]=2W[c]}}}, and {{{H[l]=2H[c]}}}



{{{SA=2(4L[c]*W[c]+4L[c]*H[c]+4W[c]*H[c])}}} Multiply



{{{SA=8(L[c]*W[c]+L[c]*H[c]+W[c]*H[c])}}} Factor out the GCF 4 and multiply it by the outer term 2



So the surface area of Lucy's box is 8 times the surface area of Charlie's box. So she needs 8 times more paint than Charlie.



So simply multiply the amount of paint that Charlie uses (one half gallons) by 8 to get


{{{8(1/2)=8/2=4}}}


So she needs 4 gallons of paint to cover her box.