Need help! Calculus - Optimization box question

[hachiebang]

Hello i am having trouble with this Grade 12 optimization question. Can someone either write this in paint or something so i can see every step clearly? thanks.

If 2700cm^2 of material is available to make a box with a square base and open top, find the dimensions of the box with maximum volume. What is the maximum volume?

(And yeah, it 2700 cm squared, not cubed like volume)

I also attached the pic of how the box is drawn from the question.

 

[Unco]

Standard practise: form an equation for the area, solve for one variable (h will be easier here) and substitute this into the equation for the volume (which you need to form).

After having volume in terms of one variable, maximise (differentiate, set to zero, solve for x).

Check the second derivative to ensure your maximum is indeed a maximum.

Calculate the volume.

 

[galactus]

This problem isn't that bad. They're giving you the surface area required. You need to maximize the volume.

Count up the surface area. The base clearly has area \(\displaystyle x^{2}\), since it's square. Right?.

Each side has area \(\displaystyle xh\). See?. There are 4, so we have \(\displaystyle 4xh\) as the area of the 4 sides. Total surface area is then \(\displaystyle x^{2}+4xh=2700\). See?.

The volume is given by \(\displaystyle V=x^{2}h\)

Now, solve the surface area equation for h, sub it into the volume equation, differentiate, set to 0 and solve for x. There you have it.

 

[hachiebang]

This problem isn't that bad. They're giving you the surface area required. You need to maximize the volume.

Count up the surface area. The base clearly has area \(\displaystyle x^{2}\), since it's square. Right?.

Each side has area \(\displaystyle xh\). See?. There are 4, so we have \(\displaystyle 4xh\) as the area of the 4 sides. Total surface area is then \(\displaystyle x^{2}+4xh=2700\). See?.

The volume is given by \(\displaystyle V=x^{2}h\)

Now, solve the surface area equation for h, sub it into the volume equation, differentiate, set to 0 and solve for x. There you have it.

So i take the Total surface area \(\displaystyle x^{2}+4xh=2700\), isolate h, then i will sub into the \(\displaystyle V=x^{2}h\), then i differentiate, then i sub in x (51.96cm)?

 

[hachiebang]

This problem isn't that bad. They're giving you the surface area required. You need to maximize the volume.

Count up the surface area. The base clearly has area \(\displaystyle x^{2}\), since it's square. Right?.

Each side has area \(\displaystyle xh\). See?. There are 4, so we have \(\displaystyle 4xh\) as the area of the 4 sides. Total surface area is then \(\displaystyle x^{2}+4xh=2700\). See?.

The volume is given by \(\displaystyle V=x^{2}h\)

Now, solve the surface area equation for h, sub it into the volume equation, differentiate, set to 0 and solve for x. There you have it.

So i take the Total surface area \(\displaystyle x^{2}+4xh=2700\), isolate h, then i will sub into the \(\displaystyle V=x^{2}h\), then i differentiate, then i sub in x (51.96cm)?

i did everything and my answer came out to be +-30cm^3, is that right anyone? Which one am i suppose to use - or + because it's a maximum question?

Is that all to it, do i have to plug that into anything?

 

[Unco]

We can't have a negative side length, so x=30 is the one to go with.

You would want to check this gives a maximum volume with the second derivative test.

The question asks for the maximum volume, so substitute x=30 into your equation for volume expressed in terms of x.