Question

Screenshot attached calculus homework :)

Answer 1

@ganeshie8 do u know how i can approach this problem?

Answer 2

With any of these optimization problems, you're pretty much always coming up with 2 equations, one which you wish to maximize and the other which you take information from. The steps I do in this problem are what you'll always be looking to try and do. It can get more or less complicated, but this is almost always the basic idea. So for you, you're given information about surface area and want to maximize volume, so that helps you out as far as 2 equations. It has a square base, so we know 2 of the 3 dimensions are the same as we set up the two equations.

So surface area first. So these rectangular solids have 6 sides. The top and bottom will be the same area, while the other 4 sides are equal to each other, but different than the top and bottom. For the top and bottom, since they have the same dimensions, we can say their area is each \(x^{2}\). The other 4 sides we can say each have an area of \(xy\), since the 3rd dimension isnt the same. Those together make a surface area of
\(2x^{2}\ + 4xy\ = 181.5\)

The volume we can just write as:

\(x^{2}y\)

So that's 2 equations. Next step is substitution. We want to solve for a variable in our known equation and substitute it into the equation we wish to maximize. The only variable we can really isolate in the surface area equation is y, so we can solve for y.

\(2x^{2}\ + 4xy\ = 181.5\ \implies\ 4xy = 181.5 -2x^{2}\ \implies\ y=\frac{181.5-2x^{2}}{4x}\)

So this value of y is substituted into the equation we wish to maximize. So that substitution gives us:
\(\frac{x^{2}(181.5-2x^{2})}{4x}\ = \frac{181.5x-2x^{3}}{4}\)

After this substitution, we want to take the derivative of what we have and set it equal to 0. You may recognize this as finding critical points (which help you find mins and maxes), and it is. So if we take this derivative and set it equal to 0, we have:
\[\frac{ 181.5-6x^{2} }{ 4 }=0\] If you solve for x here, you would get x = 5.5 and -5.5 (I can show those steps if you wish after). Now, we're talking about dimension here, so a negative answer isn't possible. So this gives us a critical point of 5.5 and only 5.5. But now that we have that number, we can go back to our surface area equation and get the other dimension of our figure.

So using this 5.5 value for x, we have:
\[2(5.5)^{2} + 4(5.5)y = 181.5 \implies y = 5.5\]And this answer makes sense. If you have a rectangular solid, its maximum volume will occur when you have a perfect cube, so we should be getting the same answer for all the dimensions. So each side of this solid will be 5.5cm. Hope all this makes sense :)

Answer 3

I just saw this late and didn't notice the answer and submitted my homework, but WOW!! this makes a lot of sense, thank you so much!! I really appreciate it, i read it over and over, && it makes perfect sense!!! :D i understand it, i will screen shot this and save it to my laptop, this will be very helpful:)