Your car has no cup holder, so you must place your filled coffee cup on the passenger seat beside you when you start out in the morning. Bitter experience has taught you that the cup is least stable and most prone to spill when it is completely full, but becomes more stable as you drink the coffee and thereby lower its level in the cup.

The picture on the left shows a coffee cup partially filled with coffee. We will assume that it is the most stable when the centroid of the cup-plus-coffee is lowest.

Assume that the centroid of the cup-plus-coffee is represented by the function of the depth y (in cm) of the coffee in the cup:

f ( y ) = 7 ( 3 + 0. [ c ] ) 2 64 + y 2 2 ( 3 + 0. [ c ] ) 2 16 + 2 y where 0 y H

Calculate the optimal depth y of the coffee in your cup. (HINT: the centroid function is lowest when f ( y ) = 0 .) Round your answer to the nearest whole number. Include the sketch of the centroid function f ( y ) and verify your solution graphically.