Optimization Problem about a Lean-To

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SUMMARY

The discussion focuses on optimizing the dimensions of a lean-to structure with a wooden floor and a roof sloping at a 45-degree angle, using a total of 300 square feet of wood. The floor area is defined as a rectangle with dimensions x and y, leading to the equation for the area as xy. The challenge lies in establishing a relationship between x and y, particularly in calculating the height difference between the front and back due to the slanted roof, which affects the overall floor area. Participants seek assistance in deriving the necessary equations to maximize the floor area while adhering to the wood constraints.

PREREQUISITES
  • Understanding of basic geometry, particularly rectangles and trapezoids.
  • Familiarity with optimization problems in calculus.
  • Knowledge of how to express area as a function of variables.
  • Ability to visualize geometric relationships in three dimensions.
NEXT STEPS
  • Study optimization techniques in calculus, focusing on maximizing area under constraints.
  • Learn how to derive equations from geometric properties, particularly for slanted surfaces.
  • Explore the concept of slant height in triangular and trapezoidal shapes.
  • Investigate practical applications of wood usage in construction and design optimization.
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Students and professionals in architecture, engineering, and mathematics who are interested in optimization problems and geometric design principles.

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Here is what what written about this Lean-To:
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?

|\
|45degrees in this top corner
| \
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.
 
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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.
 

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