Well, they do manufacture storage tanks with cylindrical tops and cone-shaped bottoms. Ask your students why, then share the following excerpt 'borrowed' from the website of a company which makes these:
"Cone bottoms provide for quick and complete drainage."
Alright already - enough motivation for a geometry problem! No calculus needed!
A conical storage tank with a maximum depth of 10 feet is completely filled with a chemical solution. Some of its contents are then drained from the bottom.
Ask your students:
(a) When depth of liquid falls to 5 ft, explain intuitively (no calculations) why much more than half the contents has drained out.
(b) Now for the geometry application...
What % of the total liquid has been drained when depth drops to 5 ft?
(c) (More challenging) What should depth be for tank to be half full? Give both one place approx and 'exact' answer.
Ans: approx 7.9 ft
I'll leave exact answer to my astute readers!
Note for instructor: You may want to explore different depths like 6', 7', 8' first to see how close we can come to half full.
QUESTIONS FOR THE INSTRUCTOR
WHAT ARE THE BIG IDEAS HERE?
DO YOU BELIEVE THIS CONCEPT IS ASSESSED ON SATs?
GIVE PRECISE WORDING OF THIS OBJECTIVE IN THE CORE CURRICULUM.
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