Rectangular Playhouse

[debased]

I'm stuck on an applied optimization problem about minimizing cost and finding the dimensions of the playhouse. I'll post what I have and the problem below:

A rectangular playhouse is to be constructed with 6 foot sidewalls and covered with a flat roof. The front wall of the playhouse will cost 12$/ft^2, the other 3 walls will cost 8$/ft^2, and the roof will cost 6$/ft^2. The playhouse must have a floor area of 100ft^2. Find the dimensions of the playhouse that minimize cost and find that cost.

So I'm fairly certain I set up the cost function and constraint wrong, even though I'm pretty sure the answer is x = 10ft.

x^2 = 100ft^2

C(x) = 6x^2 + 216x

Obviously differentiating that gives me an x that doesn't satisfy the constraint. Where did I go wrong?

 

[Deleted member 4993]

I'm stuck on an applied optimization problem about minimizing cost and finding the dimensions of the playhouse. I'll post what I have and the problem below:

A rectangular playhouse is to be constructed with 6 foot sidewalls and covered with a flat roof. The front wall of the playhouse will cost 12$/ft^2, the other 3 walls will cost 8$/ft^2, and the roof will cost 6$/ft^2. The playhouse must have a floor area of 100ft^2. Find the dimensions of the playhouse that minimize cost and find that cost.

So I'm fairly certain I set up the cost function and constraint wrong, even though I'm pretty sure the answer is x = 10ft.

x^2 = 100ft^2

C(x) = 6x^2 + 216x

Obviously differentiating that gives me an x that doesn't satisfy the constraint. Where did I go wrong?

Start by assuming:

Length of play house = L

Width of play house = W

I am assuming that the height of the play-house is 6'.

So

cost = 12*(L*6) + 8 * (L*6) + 2* 8 * (W*6) + 6 * (L*W)

Constraint:

L* W = 100

Continue....

 

[Dr.Peterson]

I'm stuck on an applied optimization problem about minimizing cost and finding the dimensions of the playhouse. I'll post what I have and the problem below:

A rectangular playhouse is to be constructed with 6 foot sidewalls and covered with a flat roof. The front wall of the playhouse will cost 12$/ft^2, the other 3 walls will cost 8$/ft^2, and the roof will cost 6$/ft^2. The playhouse must have a floor area of 100ft^2. Find the dimensions of the playhouse that minimize cost and find that cost.

So I'm fairly certain I set up the cost function and constraint wrong, even though I'm pretty sure the answer is x = 10ft.

x^2 = 100ft^2

C(x) = 6x^2 + 216x

Obviously differentiating that gives me an x that doesn't satisfy the constraint. Where did I go wrong?

This is meaningless without information about what x means! What is it?

If I am guessing correctly, you assumed that the floor is square. That leaves nothing to vary!

Try making it a rectangle.

 

[debased]

Start by assuming:

Length of play house = L

Width of play house = W

I am assuming that the height of the play-house is 6'.

So

cost = 12*(L*6) + 8 * (L*6) + 2* 8 * (W*6) + 6 * (L*W)

Constraint:

L* W = 100

Continue....

Ok so I assumed the roof and the floor were squares for some reason. Going off that I get to

C(L) = 12*(L*6) + 8 * (L*6) + 2* 8 * (W*6) + 6 * (L*W) = 120L + 9600L^-1 + 600

Then C prime of L = -9600L^-2 + 120

Set that to 0 to find the stationary points and I get

L = 4 * sqrt(5)

So then W = 100/(4 * sqrt(5)) = 91444/8179

And the cost is C(4*sqrt(5)) = 2,746.63$


Did you get the same answer?