| Alma College has a rectangular patch of grass measuring 55 yards by 40 yards. Suppose the college wants to construct a diagonal sidewalk one yard wide exactly in the manner indicated in the diagram. Assuming that the sidewalk is 6 inches deep, precisely how many cubic yards of cement would be required? [Note that the drawing below is not to scale, and the sidewalk has been intentionally drawn wider than one yard for illustrative purposes.] |  |
Solution by Jon Dent, Jenison High School
Since 
,
it follows that 
.
Thus, by the Pythagorean theorem, we have
Using the quadratic formula, we get a positive root of:
The sidewalk has the shape of a parallelogram, so its area is
equal to 
. Thus,
the volume
of the sidewalk is:
cubic yards.
Also solved by: Yu Han, Shelly Jiang (Plymouth-Canton Ed. Park); Gerald Chang (Troy Athens H.S.); Emily Auperlee, Adam Carlson, Michelle DeBruyn, Steve Gordon, Rebecca Lindsay, Mike Meindertsma, Elise Miller, Rob Reyers, Nick Schultz (Forest Hills Northern H.S.); Alex Bordyukov, Hyunje Grace Cho (Grosse Pointe South H.S.); Sunnil Abraham (Adrian H.S.); Keith Boruta (Plymouth Christian Academy); Chris Szybisty (Detroit Catholic Central H.S.); Joe Tsao (Rochester H.S.); Alexander Tang (Ann Arbor Huron H.S.); Ramsey Gilbertsen (Marquette H.S.); Miles Kaufmann (Traverse City Central H.S.); Trey Springer (Grand Rapids Catholic Central H.S.); Kyle DeBoer (Forest Hills Eastern H.S.); Husam Alghanem (Grand Blanc H.S.); Derek Elenbaas (Canton); and Katherine Strong (Grand Rapids).