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anonymous
 5 years ago
An open box is being constructed whose base length is 3 times the base width and whose volume is 50 cubic meters. If thematerials used to build the box cost $10 per square meter for the bottom and $6 per square meter for the sides, what are the dimensions for the least expensive box?
Help?!?! I feel as if not enough information is provided. Yet again I could be wrong? How do I solve this problem?
anonymous
 5 years ago
An open box is being constructed whose base length is 3 times the base width and whose volume is 50 cubic meters. If thematerials used to build the box cost $10 per square meter for the bottom and $6 per square meter for the sides, what are the dimensions for the least expensive box? Help?!?! I feel as if not enough information is provided. Yet again I could be wrong? How do I solve this problem?

This Question is Closed

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0this is an optimization problem, did you see that in class?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Yes I do recognize that its an optimization problem, in which we have to alter our formulas.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0do you need an answer?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0setting up and understanding, please.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0okay, so first lets take care of the constraint. i.e. the volume of this box is 50 cm^3

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0how about \[V=3x^2y\] and we know that \[50=3x^2y\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0where did the 3x^2 come from?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0woah sorry i had to go fix a computer ill let ebbflo take over hes got a good flow going,

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0the length is 3 time the width

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0so 3x^2 is the area of the base and I just let y be the heigth of the box

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0okay now we need the surface area formula for this box which since there is no top is \[S=3x(x)+2(3x)y+2xy\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Area = base * height so the base =3x and the height=x

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0i am just adding up the areas of the sides of the box

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0the base of the box has dimensions (3x)cm by (x)cm

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0are you good with the surface area formula I presented?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0If so We can proceed to the cost function

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0can you explain to me how you arrived to your Surface area

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0since this box only has 5 sides we just add up the areas of those sides the area of the bottom is (3x)(x) the combined area of one of the pairs of equal sides is 2(3x)(y) and the combined area of the remaining pair of equal sides is 2(x)(y) then I just summed those up

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0so that's where I got my formula for the surface area

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0so now ti get the cost function we just multiply 10 by the area of the bottom and 6 times each of the combined areas of the sides like so \[C(x,y)=10(3x^2)+6(6xy)+6(3xy)\] collecting like terms we pretty it up to get \[C(x,y)=30x^2+48xy\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0sorry I made a slight mistake in that formula

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0the last one I wrote was correct

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[C(x,y)=10(3x^2)+6(6xy)+6(2xy)\] should have been the first one from which I got \[C(x,y)=30x^2+48xy\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0now we are ready to write the cost function above as a function of x only by way of the constraint

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0so since \[50=3x^2y\]\[y=\frac{50}{3x^2}\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0so \[C(x)=30x^2+48x(\frac{50}{3x^2})\] and cleaning it up we get \[C(x)=30x^2+\frac{800}{x}\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0if we are good, then now it is time to derive

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[C^\prime(x)=60x\frac{800}{x^2}\] get a common denominator to make finding the critical points easier we get\[C^\prime(x)=\frac{60x^3800}{x^2}\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0since this is a box with known volume we know \[x\neq 0\] so just solve \[60x^3800=0\] for x

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0we get \[x=2(\frac{5}{3})^\frac{1}{3}\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0this is the only value we get so many people quit there and just go with but you can confirm this is a minimum value via the second derivative test for max/min

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0use that vale to get the other dimensions and you are done...

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0glad it helps, this is easier to explain with the aid of a sketch

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0I was just looking over the work and you told me to use my x value to figure out my other dimensions. Do i do that by taking the second derivative?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0no just find the height by plugging that x value into the constraint we solved for why and the other dimension is just 3 time that x value we got
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