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

- anonymous

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- anonymous

@Loser66

- triciaal

|dw:1440636145968:dw|

- triciaal

|dw:1440636232302:dw|

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## More answers

- triciaal

volume changes with area and area changes with height

- anonymous

okay, from there can I just solve for the h and then plug it into the volume function which is V=s^2*h

- anonymous

|dw:1440636504171:dw|

- triciaal

@jim_thompson5910

- jim_thompson5910

solve for h to get
\[\Large 1200 = s^2 + 4sh\]
\[\Large 1200 - s^2 = 4sh\]
\[\Large 4sh = 1200 - s^2\]
\[\Large h = \frac{1200 - s^2}{4s}\]

- jim_thompson5910

\[\Large V = s^2*h\]
\[\Large V = s^2*\frac{1200 - s^2}{4s}\]
\[\Large V = s*\frac{1200 - s^2}{4}\]
\[\Large V = \frac{1200s - s^3}{4}\]
so it looks good so far

- anonymous

the volume formula would be

- anonymous

|dw:1440637090561:dw|

- anonymous

right?

- triciaal

yes read above by jim he found the expression for h then substitute in the formula for the volume

- jim_thompson5910

yes correct, the volume is s^2*h

- jim_thompson5910

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

- anonymous

What intervals should I use, My graph doesn't show it clearly.

- jim_thompson5910

https://www.desmos.com/calculator/bbcpea5fld
my window I used was
xmin = -10
xmax = 40
ymin = -3000
ymax = 5000

- jim_thompson5910

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

- anonymous

is it 4000

- jim_thompson5910

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

- anonymous

OMG, thank you so much! ... DO you think you can help me with another one?

- jim_thompson5910

sure

- anonymous

provide one the two positive integers whose sum is 200 and whose product is a product

- triciaal

what do you mean the product is a product?

- anonymous

like if the two integers are multiplied, that equals to the maximun value

- triciaal

the constant is not prime?

- anonymous

I don't know to be honest

- jim_thompson5910

do you mean
`provide one the two positive integers whose sum is 200 and whose product is a MAXIMUM` ??

- anonymous

yes

- jim_thompson5910

`integers whose sum is 200`
x+y = 200
solve for y to get
y = 200-x

- jim_thompson5910

the product is the result of multiplying
x*y = x*(200-x) = 200x - x^2

- jim_thompson5910

the goal is to max out 200x - x^2

- jim_thompson5910

you can graph 200x - x^2 or complete the square to find the vertex

- anonymous

I got (100, 10000)

- anonymous

@jim_thompson5910 so the answer is 10000

- jim_thompson5910

so the two numbers are 100 and 100
100+100 = 200
100*100 = 10,000
the max product is indeed 10,000

- jim_thompson5910

I think they are asking for one of the two numbers and not the max product

- anonymous

okay, thank you

- jim_thompson5910

sure thing

- triciaal

@jim_thompson5910 thanks

- jim_thompson5910

you're welcome

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