Finding maximum Volume provided $10,000 to spend on material to build a Green House.

[SinaSyndrome]

• The Green house needs to be rectangular in shape.

• The floor dimensions must be in a 3:2 ratio of length.

Here is the cost for the material being used to build the Green House:

Material	Used for	Cost per ft^2
Special Coated Glass	Front	$4.00
Regular Glass	Sides & Back	$3.00
Thick Tempered Glass	Roof	$10.00
Concrete	Floor	$5.00

_______________________________________________________________________________________
Let V = volume | l = length | w = width | h = height | A = area

V = l * w * h

A = V/h = l * w
_______________________________________________________________________________________

Given that the floor dimensions must be in a 3:2 ratio, the volume equation becomes:

V = 3x * 2x * h

V = 6x^2 * h
_______________________________________________________________________________________

Now I know the cost equation is,

C = 30x^2 + 60x^2 + 12xh +12xh +9xh

C = 90x^2 + 33xh

Since we know the cost we can say,

10,000 = 90x^2 + 33xh
_______________________________________________________________________________________
_______________________________________________________________________________________

This is where I get lost!

I think I need to use the equation, 10,000 = 90x^2 + 33xh,
to find the value of h

Once I find h I believe I need to plug it into the equation,

V = 6x^2 * h, to then solve for Volume

I'm not really sure if I'm doing this right up to here! I feel like I'm missing something, possibly the x value in the volume equation?

Sorry for the super long post. I gave all the information that I believe is relevant, but if you think I'm missing something please let me know and I'll try to provide more.

 

[Romsek]

<snip>

Now I know the cost equation is,

C = 30x^2 + 60x^2 + 12xh +12xh +9xh

C = 90x^2 + 33xh

Since we know the cost we can say,

10,000 = 90x^2 + 33xh
_______________________________________________________________________________________
_______________________________________________________________________________________

This is where I get lost!

I think I need to use the equation, 10,000 = 90x^2 + 33xh,
to find the value of h

Once I find h I believe I need to plug it into the equation,

V = 6x^2 * h, to then solve for Volume

I'm not really sure if I'm doing this right up to here! I feel like I'm missing something, possibly the x value in the volume equation?

Sorry for the super long post. I gave all the information that I believe is relevant, but if you think I'm missing something please let me know and I'll try to provide more.

Have you learned about Lagrange multipliers yet?

If not you are on the right track above. You want to solve the Cost equation for x in terms of h. Then plug that x(h) into the volume equation to get V = 6(x(h))² * h. Then find dV/dh and set to 0 and solve for h. x(h) then gives you x. l = 3x, w = 2x.

The easier way is to use a Lagrange multiplier \(\displaystyle \lambda\).
First create F(x,h,\(\displaystyle \lambda\)) = Volume(x,h) - \(\displaystyle \lambda\)(Cost(x,h)-10000)
Take the gradient of F, set each component to 0, and solve for {x,h,\(\displaystyle \lambda\)}.

I'll let you work through the details.

 

[SinaSyndrome]

Have you learned about Lagrange multipliers yet?

If not you are on the right track above. You want to solve the Cost equation for x in terms of h. Then plug that x(h) into the volume equation to get V = 6(x(h))² * h. Then find dV/dh and set to 0 and solve for h. x(h) then gives you x. l = 3x, w = 2x.

The easier way is to use a Lagrange multiplier \(\displaystyle \lambda\).
First create F(x,h,\(\displaystyle \lambda\)) = Volume(x,h) - \(\displaystyle \lambda\)(Cost(x,h)-10000)
Take the gradient of F, set each component to 0, and solve for {x,h,\(\displaystyle \lambda\)}.

I'll let you work through the details.

I've never learned about Lagrange multipliers before, sorry.

So I found h


h = (10000 - 90x² )/(33x)
__________________________________________________
then I plugged that in for h in the Volume function. Giving me,


V = 6x² ((10000-90x² )/(33x))


Then found the derivative of that function,


V' = -(20/11) * (27x² - 1000)


Setting the derivative function to 0, it gave me the x value of,


x = 6.0858
__________________________________________________
Plugging that value into the original Volume function gave me,


V = 6(6.0858)² * ((10000-90(6.0858)² )/(33(6.0858)))


V = 7376.7348
___________________________________________________
Does that mean the maximum volume would be 7376.7348 ft³​?

 

[Romsek]

I've never learned about Lagrange multipliers before, sorry.

So I found h


h = (10000 - 90x² )/(33x)
__________________________________________________
then I plugged that in for h in the Volume function. Giving me,


V = 6x² ((10000-90x² )/(33x))


Then found the derivative of that function,


V' = -(20/11) * (27x² - 1000)


Setting the derivative function to 0, it gave me the x value of,


x = 6.0858
__________________________________________________
Plugging that value into the original Volume function gave me,


V = 6(6.0858)² * ((10000-90(6.0858)² )/(33(6.0858)))


V = 7376.7348
___________________________________________________
Does that mean the maximum volume would be 7376.7348 ft³​?

That's the right answer but you have to be a bit more careful and check the 2nd derivative at x is negative to ensure a maximum.
Note V' = 0 has a 2nd root. You might want to look at that and convince yourself it can be ignored.

btw soving for h in terms of x turned out to be much easier than vice versa so I'm glad you did it that way.

 

[SinaSyndrome]

That's the right answer but you have to be a bit more careful and check the 2nd derivative at x is negative to ensure a maximum.
Note V' = 0 has a 2nd root. You might want to look at that and convince yourself it can be ignored.

btw soving for h in terms of x turned out to be much easier than vice versa so I'm glad you did it that way.

Awesome, thanks for all the help!

Yea I just figured I needed to use the positive root. I knew there was a negative one, but I assumed the negative one was irrelevant for this application.