Building a scale-model doghouse

Some doghouses are shaped like small barns. Barns usually have gambrel roofs—a gambrel roof is a roof where each side has two slopes, a steeper lower and a flatter upper one.  In this unit we will build a miniature doghouse with a gambrel roof.

We found some miniature toy dogs.

(You can find them on the internet, for example, at http://www.toyconnection.com/Merchant2/merchant.mvc?Screen=CTGY&Category_Code=A&qts=google&qtk=plastic%20toy%20dogs.)

Each student will get a little dog and will build a doghouse that is scaled for the dog.  This is how it will look.

We made ours without a floor, but you may include a floor if you want.

Supplies.

Miniature toy dog, half-centimeter grid graph paper (you can find this on the internet, for example, at http://www.mhhe.com/math/ltbmath/bennett_nelson/conceptual/instructor/grids/HalfCentGrid.pdf), poster board or heavy cardstock, ruler, compass, scissors, scotch tape, protractor or index card, calculator (a 4-operation calculator is sufficient for tasks 1 and 2, but a scientific calculator is needed for task 3.).

On the half-centimeter grid graph paper, lay out the plan below with dimensions as given for the front wall.

It can be laid out in a 9 cm by 7 cm rectangle.

The plan for the back wall is identical, but of course without the door. (Each one is a seven-sided polygon.)

Copy the two plans onto your poster board.  You can do this by laying the graph paper over the poster board, fastening it with paper clips, and using the point of your compass to make holes through the paper and poster board at the vertices of the polygon.  Then connect the vertices with straight segments.

The floor is 9 cm by 8 cm.  We will make our roof and side walls from a rectangular piece, which will have five folds.  One side of this rectangle will be the length of the doghouse, or 8 cm.  What is the length of the other side? See the picture.  Use the Pythagorean theorem to compute the lengths of the sloped parts of the roof. You can make the rectangle first on the graph paper, and copy it onto the poster board, or make it directly onto the poster board. Be sure the corners are right angles!  (Use a protractor or index card to check.)

The rectangle forming the roof is 8 cm by 18.6 cm.  Lay it out.  Draw straight segments where the folds will be: at 3 cm, 5.9 cm, 9.6 cm (the middle), 12.6 cm, and 15.5 cm. Using the point of your compass, score along the lines so that your folds will be sharp.

Cut out the pieces and carefully construct the doghouse.  Now you have a home for your little doggie!

Find the volume of the doghouse.

It can be viewed as a prism with a 7-sided base and with height 8 cm.   The volume of a prism is (area of base)*height.  Here is one way to find the area of the base (the front or back of the doghouse). We first inscribe it in a 9 cm by 7 cm rectangle.  We will find the area of the portion of the rectangle outside the figure, and subtract that amount from 63 sq cm.  (See picture.)

Area of square A = 2.25 sq cm

Area of triangle B = ½*3*1½ = 2.25 sq cm

Area of triangle C = ½*1½ *2½ = 1.875 sq cm

So area of base = 63- 2*(2.25+2.25+1.875) =  50.25 sq cm

Volume of doghouse = 50.25*8 = 402 cu cm

Since there are 2.54 cm in one inch, 402 cu cm is 402/(2.543)= 24.5 cu inches.

Design a doghouse similar to the one you have just built, but with volume half as big.

Hint.

The new volume will be half of 402, or 201 cu cm.

To find the new dimensions, each linear dimension is multiplied by the cube root of ½, or .793700526.  So the front of the doghouse will have width 3√(.5)*9 cm ≈ 7.15 cm.

Here is a picture of the two dog houses, one with twice the volume as the other.

You may check the volumes using rice.