# Optimization Problem about a Lean-To

1. Nov 24, 2007

### xcolleenx

A lean-to has a wooden floor, a 10 feet high open front, wooden side and back walls, and a wooden roof that tilts down at an angle of 45 degrees. What are the dimensions of the lean-to with the largest possible floor area that can be constructed with 300 square feet of wood?

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|45degrees in this top corner
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10ft | \
| \
| |
front |____| back

Tried to draw a picture...

Ok the first thing i need to do is express the floor area as a function of two variables. I know this. The floor is a simple rectangle so I will call the area xy. Next i need to "find an equation relating the two variables x and y." This is where I am having trouble. A hint was given that reads: "The roof is still a rectangle. You will need to find how much lower the back is than the front. You will also need to find the "slant height" of the roof. The sides are trapezoids. You may want to view them as a triangle on top of a rectangle."

I am having trouble doing this second part. If i can get help with this i solve the rest of this problem. Thank you for any help.

2. Nov 24, 2007

### mgb_phys

Simply need an equation that describes the projected floor area of the roof.
This is simply the horizontal distance from the wall to the edge of the roof * the width.
The floor area is this distance * width.

You know the area of the roof and how much wood it uses, and the area of the floor.
As you make the roof longer you use up wood for the roof faster than you gain floor area.