MATH OLYMPIAD (1998) Session I Problem

GRADE 5: The Greenhouse Problem

You are visiting your uncle who owns a nursery. He has 100 boards to build the walls of a greenhouse. He will make the roof and door out of glass himself. The boards are all 2 inches by 6 inches across (2 by 6's) and are all 8 feet long. He wants you to design his greenhouse!

  1. The inside floor area and the exterior shape must be rectangular in shape.
  2. He wants the maximum floor area you can get.
  3. Since this is a greenhouse, he wants the maximum window area you can design.
  4. The walls are to be 8 feet high.

Here are the parts to a wall:

To make sure your greenhouse is safe, there are building code rules you must follow. They are:

  1. Studs (the vertical boards) must be spaced at either 12 or at 24 inches from center to center. All studs in all walls must be placed at the same spacing.
  2. The headers (horizontal boards over window) must be thick to support the weight of the roof. A wider window or door requires a thicker header. Here's how thick they must be:
  3. Window or door width

    Header thickness

    2 feet or less

    2 inches (1 board)

    3 feet or less (greater than 2)

    4 inches (2 boards)

    4 feet or less (greater than 3)

    8 inches (4 boards)

    5 feet or less (greater than 4)

    10 inches (5 boards)

  4. Every wall must have a base plate and a top plate each exactly 2 boards thick.
  5. Every window requires a base plate (1 board) and a header, and must not be more than 5 feet wide.
  6. Windows must not be more than 5 feet high.
  7. Windows must be supported on both ends by studs, as shown in the figure.



Design the walls of your uncle's greenhouse. Use the GREENHOUSE PLAN SHEET to draw neat pictures of your greenhouse. The pictures you draw should show the floor and walls of your design. Be sure to include all dimensions on your drawings. Your dimensions may be in feet and inches.

Try to meet all your uncle's requirements, stay within the building code rules, and use 100 boards or less.

Your design will be graded on how well you meet all building code rules and how well you meet your uncle's requirements. How well you meet your uncle's requirements will be measured by this quality equation:

Q = ( F + 2W )

Q = quality of your design
F = interior floor area in square feet
W = total window area in square feet
S - spacing between stud centers in inches

This equation means that the quality of your design goes up when your studs are closer together. At the bottom of the back of the GREENHOUSE PLAN SHEET is a space for you to compute your quality "Q" and tell us why your design is best.

You have 1 hour to complete this task. Good luck!





School__________________ Team number: _____________


For your design, what is:

F (the inside floor area) = ____________

W (the total window area)= ____________

S (the stud spacing) = ____________

Q (the design quality) = ____________