+101 Some more help requested for a weird question
According to USPS, a parcel can measure no more than 108 inches in length and girth combined. The length is designated to be the longest side and girth is the distance around a cross-section perpendicular to the length.
The questions that hold all the answers to life are...
1.Assuming the cross section is a square sketch the parcel???
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2. Write an equation to model the volume???
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3. What is the maximum volume???
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4. What are the dimensions of the parcel that produces the max volume???
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If possible could you show your steps and explain them because I would like to understand this as well and not just get the answer.
Thank you!!!
+37146 You have a picture.....use that as your sketch
2.volume = l x w x w = lw^2 (w= side of the square)
3. 54000 cu in is max
4. l + 4w =180 or l= 180-4w substitute this in to #2
(180-4w)(w^2) = volume
(180w^2 - 4w^3) a 'local' max will be where slope = 0
Derivative 360 w - 12w^2 = 0
w = 0 or 30 (throw out '0' obviously)
w=30 then l = 180-4w=60
volume max = l x w x w = 60 x 30 x 30 = 54000 cu in
+101 Sorry, I meant 108, not 180 my apologies. I edited it back to 108 and will try to solve it from what you gave me but I would definitely appreciate it if anyone answers the question again.
+129849 1. It looks like the picture pretty much shows the cross-section as being a square
2. We have that G + L = 108 ⇒ G = 108 - L
Assuming that the cross-section is a square....its side is 1/4 of the girth and can be expressed as:
G/4 = (108 - L) / 4
So....the volume can be expressed as L [ (108 - L) / 4] ^2
So we have that
V = L [ (108 - L) / 4]^2 = L [ 27 - L/4]^2 = L [ L^2/16 - 27L/2 + 729 ] =
L^3/16 - 27L^2//2 + 729L
To find the max volume, we can use some Calculus or a graph
The graph seems easiest....here it is :
https://www.desmos.com/calculator/ybfbga97jl
3. Looking at the graph, the maximum volume is [ again, assuming a square base ] = 11664 in^3
4. The Length, L = 36 in
The Girth, G = 108 - 36 = 72 in
So....the side of the square is 1/4 of this = 18 in
So...the dimensions are 18 in x 18 in x 36 in = 11664 in^3
Note that the restrictions have been met Girth + Length = (4 * 18) + 36 = 72 + 36 = 108 in
