Guys please help me, I give medals If 1,200 cm2 of material is available to make a box with a square base and an open top, find the maximum volume of the box in cubic centimeters

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Guys please help me, I give medals If 1,200 cm2 of material is available to make a box with a square base and an open top, find the maximum volume of the box in cubic centimeters

Mathematics
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Other answers:

volume changes with area and area changes with height
okay, from there can I just solve for the h and then plug it into the volume function which is V=s^2*h
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solve for h to get \[\Large 1200 = s^2 + 4sh\] \[\Large 1200 - s^2 = 4sh\] \[\Large 4sh = 1200 - s^2\] \[\Large h = \frac{1200 - s^2}{4s}\]
\[\Large V = s^2*h\] \[\Large V = s^2*\frac{1200 - s^2}{4s}\] \[\Large V = s*\frac{1200 - s^2}{4}\] \[\Large V = \frac{1200s - s^3}{4}\] so it looks good so far
the volume formula would be
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right?
yes read above by jim he found the expression for h then substitute in the formula for the volume
yes correct, the volume is s^2*h
you want to find the max volume, so you'll need to graph \[\Large y = \frac{1200x-x^3}{4}\] and locate the local max
What intervals should I use, My graph doesn't show it clearly.
https://www.desmos.com/calculator/bbcpea5fld my window I used was xmin = -10 xmax = 40 ymin = -3000 ymax = 5000
on the desmos graph, you should be able to click the local max (maybe click twice) to have the coordinates of that point show up
is it 4000
yeah that local max is (20,4000) meaning that x = 20 and y = 4000 so if the side length is s = 20 cm then the max volume is 4000 cubic cm
OMG, thank you so much! ... DO you think you can help me with another one?
sure
provide one the two positive integers whose sum is 200 and whose product is a product
what do you mean the product is a product?
like if the two integers are multiplied, that equals to the maximun value
the constant is not prime?
I don't know to be honest
do you mean `provide one the two positive integers whose sum is 200 and whose product is a MAXIMUM` ??
yes
`integers whose sum is 200` x+y = 200 solve for y to get y = 200-x
the product is the result of multiplying x*y = x*(200-x) = 200x - x^2
the goal is to max out 200x - x^2
you can graph 200x - x^2 or complete the square to find the vertex
I got (100, 10000)
@jim_thompson5910 so the answer is 10000
so the two numbers are 100 and 100 100+100 = 200 100*100 = 10,000 the max product is indeed 10,000
I think they are asking for one of the two numbers and not the max product
okay, thank you
sure thing
you're welcome

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