Elysian Shadows

Posted: **Sun Aug 09, 2009 9:50 pm**

Hey guys,

I'm on my first semester of calculus (again) and were doing optimizations.

I came up with this equation for the area of a swimming pool and 4 decks surrounding the pool:

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Area = 1800

1800 = L*W*H + 10L + 20W

No problem, I chose to solve for H:

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1800 = L*W*H + 10L + 20W

1800 - 10L - 20W = L*W*H

(1800 - 10L - 20W) / (LW) = H          // cool I solved for H

Now I go back to solve for L and W:

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1800 = L*W*H + 10L + 20W

1800 = L*W( (1800 - 10L - 20W) / (LW) ) + 10L + 20W     // Substitute in H

Here's my problem, everything cancels out. 0 = 0.

I feel like I am missing something basic from algebra. This doesn't happen every time I use substitution to solve a problem, only some of the time.

Does anyone know what I am doing wrong?

note: I thought the code boxes looked nicer then the Quote boxes, but this is not actual code.

Posted: **Sun Aug 09, 2009 10:25 pm**

What you are doing isn't legal. You're trying to solve a single equation for multiple variables. There is not enough information in one equation to solve for any more than one variable.

If you need to find x variables, then you MUST have x equations.

For what you are looking for, you either 1) need to find the variable from another equation or 2) didn't do something correctly.

Posted: **Sun Aug 09, 2009 10:36 pm**

Allright, thanks. That explains that.

I think I need to use implicit differentiation to solve this actually. I think.

The problem says:

A rectangular swimming pool is to be built with an area of 1800 square feet. The owner wants 5-foot wide decks along either side and 10-foot wide decks at the two ends. Find the dimensions of the smallest piece of property which the pool can be built satisfying these conditions.

From what I can picture is the question is stating the pool is a cube with 4 decks surrounding the perimeter. Two decks being 5*L each, and two decks being 10*W each. The question says the total area = 1800, so I came up with the following equation:

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1800 = L*W*H + 10L + 20W

My guess, at this point, is somehow I need to solve L in terms of W. But that would mean I have to many variables, and possibly I could assume that H isn't a valid variable. Which doesn't make sense unless my original assumption, the shape of the pool, isn't 3dimensional.

epiphany: If the problem states the rectangle is 1800 square feet, does that mean we can ignore the depth of the pool when coming up with an equation and just use L and W?

Posted: **Sun Aug 09, 2009 10:44 pm**

Your epiphany is correct. It's talking about area, which is 2D. You can ignore height.

I'm trying to remember how to do these, though. It's been a looong time since Cal A.

Posted: **Sun Aug 09, 2009 11:07 pm**

short0014 wrote:Allright, thanks. That explains that.

I think I need to use implicit differentiation to solve this actually. I think.

The problem says:
A rectangular swimming pool is to be built with an area of 1800 square feet. The owner wants 5-foot wide decks along either side and 10-foot wide decks at the two ends. Find the dimensions of the smallest piece of property which the pool can be built satisfying these conditions.
From what I can picture is the question is stating the pool is a cube with 4 decks surrounding the perimeter. Two decks being 5*L each, and two decks being 10*W each. The question says the total area = 1800, so I came up with the following equation:
Code: Select all
1800 = L*W*H + 10L + 20W
My guess, at this point, is somehow I need to solve L in terms of W. But that would mean I have to many variables, and possibly I could assume that H isn't a valid variable. Which doesn't make sense unless my original assumption, the shape of the pool, isn't 3dimensional.

epiphany: If the problem states the rectangle is 1800 square feet, does that mean we can ignore the depth of the pool when coming up with an equation and just use L and W?

this might not be right so just forgive me if it isent. i dont have pen and paper.
anyways.

i'll assume length and width of the pool at x and y respectivly

so we know that x*y = 1800
so lets get y interms of x
y = 1800/x

now if you think about it a little you can reduce the area of the deck to this

area deck = 10*(x+10)*2 + (y*5)*2

now replace for y

area deck = 10*(x+10)*2 + ((1800/x)*5)*2
simplify

20x + 200 + 18000/x

diferentiate

(20*x^2-18000)/x^2

now find the roots, im not sure. but 30 looks like one.
so you know x = 30 is the one var.

y = 1800/x
y = 60

so w and l = 30 and 60

you should be able to take it form here.

Posted: **Sun Aug 09, 2009 11:08 pm**

Just out of curiosity what did you get for the answer? (I got 4800 sq ft, but I've never taken Calc so I dunno how you're supposed to solve this).

Posted: **Sun Aug 09, 2009 11:12 pm**

dandymcgee wrote:Just out of curiosity what did you get for the answer? (I got 4800 sq ft, but I've never taken Calc so I dunno how you're supposed to solve this).

i believe 3200 is the correct answer.

Posted: **Sun Aug 09, 2009 11:33 pm**

so we know that x*y = 1800

The question states that the pool has an area of 1800sq ft, which include 2 5*L decks and 2 10*W decks.

Do you not have to account for those when figuring out the area of the pool? I feel like L*W + the four decks should equal 1800, not just the length * width.

Am I missing something?

edit: When I draw a picture of what I believe is going on, I get a rectangle of L*W (pool), and then four extruding rectangles (the decks). The whole picture isn't a rectangle, so I don't think x*y=1800 is a correct equation.

I definately could be wrong, I'm just learning..

Posted: **Sun Aug 09, 2009 11:42 pm**

short0014 wrote:The question states that the pool has an area of 1800sq ft, which include 2 5*L decks and 2 10*W decks.

If the decks were included wouldn't the answer be 1800sq ft?

Posted: **Sun Aug 09, 2009 11:43 pm**

well its stated that the pool is built with area 1800. that means lw=1800...
there are infinitly many combinations of lw=1800

but then the question states that the owner wants to build the deck aound the pool with 5 and 10 widths respectivly, and he wants to know what the
smalls ground he can do this on, while mantaining a pool area of 1800.

no i dont know if the final area of pool and deck is rectangle, or like a plus shape.

my calculations assume that is is a rectangle.
reason being it be supid to calculate the area of the plus shape because you dont buy land like that and you still effectivly use/waste the 4 rectangles on the edges.

Posted: **Sun Aug 09, 2009 11:47 pm**

I completely agree about it being retarded, it just seems like if it were a rectangle instead of a plus shape then the mention of the deck could have been left off completely from the problem.

Posted: **Sun Aug 09, 2009 11:59 pm**

short0014 wrote:I completely agree about it being retarded, it just seems like if it were a rectangle instead of a plus shape then the mention of the deck could have been left off completely from the problem.

not really

i think they just mean a deck is being built around the pool. but on the two oppossing sides its 10 ft wide and on the other two its 5 ft

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Posted: **Mon Aug 10, 2009 12:02 am**

I'm gonna go with that. Thanks.

Solving the plus shape I'm getting 0 = 0 for solving the critical points. Thanks

Posted: **Mon Aug 10, 2009 12:10 am**

short0014 wrote:I'm gonna go with that. Thanks.

Solving the plus shape I'm getting 0 = 0 for solving the critical points. Thanks

you still should not get 0 for that.

with these min and maximization sums you allways have to differentiate.

here i'll give you another example.

assume a box gets made from 2 different meterials.
the bottom from meterial 1 costing .2 dollers a cm^2
and the sides from material 2 costing .1 dollars a cm^2

what is the beggest box in volume you can make with 1 doller.

edit. ther is no lid.

Posted: **Tue Aug 11, 2009 8:59 pm**

All right here's as far as I'm getting with your example.

When you said box, I assumed it was a square so l = w.

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Surface Area = w^2 + wh // could possibly need later

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V = (w^2) * h         // volume

h = V / (w^2)

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V(w) = (w^2) * h    // easier to follow if this is here.

V(w) = (w^2) * (V / (w^2) )     // substitute in h

V : (0, +inf) --> R // domain of V is all non-negative real numbers

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V ' (w) = (d/dw)(w^2)(h) + (w^2)(dw/dh)              // implicit differentiation, derivative of h with respect to w

V ' (w) = 2wh + (dw/dh)(w^2)                // not sure where to go after this

I got as far as I could, but I'm not told what the area is equal too, so unless you need to know what A is equal too, I am stuck.

Elysian Shadows

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