I don’t like how our book (and I assume most geometry books, at least traditional ones) breaks up similar polygons – which I think of in terms of scaling and maps – and the areas of those polygons. So I’m combining them. A couple of days after the light problem, we started a series of problems involving scaling and area.
Here was the first one, which I made up. Ensworth is another independent school in our area whose colors are orange and black.
This problem led to a very interesting and fruitful discussion both on the relationship between linear and area scale factors and the related (and commonly done wrong) idea of converting square units.
Darlene, a student and tennis player at Ensworth, has decided that she wants to pull a prank on Harpeth Hall before her big tennis match and paint their tennis courts orange. She needs to know how much paint to buy, so she uses Google Maps to print out a picture of the tennis court, as well as a picture of the Google Maps scale marker, as shown.
- Use a ruler to find the physical scale factor from the picture of the tennis courts to the ACTUAL tennis court. Your scale factor should NOT have any units in it – it should simply answer the question “How many times bigger is the real world than the picture world.” Hint: You do not need to measure the tennis courts to answer this, only the scale.
- Measure the length and width of each of the two tennis courts to one decimal place (just the part inside the lines) then use your scale factor to determine the lengths and widths of the real tennis courts.
- What is the area of the picture of each tennis court (include units, round to 2 places)?
- What is the area of the actual tennis court (include units, round to 2 places)
- If each can of paint covers 50 square meters, how many cans will Darlene need to paint both of the courts orange?
Darlene went through the steps above and found the following information (these measurement numbers are made up and will probably not match yours):
Darlene first measured the scale and found it to be 2.1 cm long. To find the scale factor, she wanted 10 m / 2.1 cm. To get the units the same, she multiplied 10 m by 100 to get 1000 cm / 2.1 cm = 476.2 . Thus she knew the real world was 476.2 times larger than the picture. She measured the length of each court to be 4.3 cm and the width to be 1.9 cm
She then calculated the area of each court to be 4.3*1.9 = 8.17 sq cm
Next, Darlene wanted to calculate the area of the real tennis court, so she multiplied the area times the scaling factor: 8.17 * 476.2 = 3,890.5 sq. cm . Since that number was large, she then divided by 100 to get an area of 38.9 sq m per court. Thus for two courts she would need 78 sq. m of paint, so she bought two cans. She wanted to move fast, so she recruited her friend Lana to help her.
Lana wasn’t convinced that two cans was enough paint, so she tried the math again. She got the same measurements and scale factor as Darlene: 4.3 cm for the length and 1.9 cm for the width, with a scale factor of 476.2 She used the scale factor to find the length of the real court to be 4.3*476.2=2,047.6 cm, or 20.5 m, and the width of the real court to be 1.9*476.2=904.8 cm or 9.05 m. She then multiplied those numbers to get an area of 9.05*20.5=185.3 sq m, meaning she would actually need 4 paint cans per court, or 8 total.
- Who is right? Or are they both wrong? You should assume their measurements make sense, and only criticize the method. Explain.