Question Coffee is dripping from a conical filter into a cylindrical coffeepot.

Question

Question

Coffee is dripping from a conical filter into a cylindrical coffeepot. Both the filter cone and the coffeepot have diameter of 6 inches. The cone has a height of 6 inches as well. When the coffee in the cone is 5 inches deep, the coffee is draining at the rate of 10in3/min.

(a). How fast is the level in the pot rising?

Hint: The volume of a cylinder of radius r and height h is V=?r2h.

(b). How fast is the level in the cone falling at the same time?

Hint: The volume of a cone of radius r and height h is V=1/3(?r2h)= ?r2h/3.

Question

(a). The profit P from selling x units of a product is given by P=15+12?x-18/x

(i). Find the average rate of change of P on the interval [1,9].

(ii). Find a formula for the marginal profit, and its numerical value when x=9

(b). A sphere of radius 3inches has its radius changing at a rate of 4 inches per second. How quickly is the volume of the sphere changing?

Explanation / Answer

The volume of a cylinder of radius r and height h is

V = ?r^2h.

olution. For the cylinder of coffee in the pot we have

dV/dt = ?r^2 dh/dt ,

(r is constant). At this particular moment we have

10 = ?

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