Because the captain of an oil tanker is so excited upon receiving the ultra-rare Regis Philbin Pokemon card in his Burger King Super Pokey Meal, he slams into the Alaskan coastline (Giant Lawsuit--I Choose You!). The oil spreads in a circle whose area increases at a constant rate of 6 miles²/hour. How fast is the radius of the spill increasing when the area is 9 miles² ?

Difficulty Rating: 
Note:    The image of Pikachu is (c)1999, Nintendo (www.pokemon.com). The use of the image does not constitute any affiliation between Nintendo of America and the Kelley AP Calculus Webpage, but I bet you probably figured that out, Mr. Smarty Pants. The image of Regis is from the cover of Entertainment Weekly, November 5, 1999 (www.ew.com). There are those of you who are probably wondering why this is written here. Those people obviously don't fear being sued as much as I do.

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Solution:

It is quite clear that we are given dA/dt in the problem (it is 6, ignoring units for now). The only problem we will have, it turns out, is finding r so that we can complete the problem. Using the information that A = 9, since the shape is a circle, we can find r in this way: So, r is approximately 1.6925687 when the area is 9. Knowing this we can actually complete the problem. (Note: you may not have realized that you needed to find r until midway through the problem--that's ok. It doesn't matter when you find r.)

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Calculus II (BC): Related Rates: The Encore Performance

        Coffee is draining from a conical filter, diameter and height both 6 inches, into a cylindrical coffee pot, diameter also 6 inches. The rate at which coffee drains from the filter into the pot is 10 in³/min.
        (a)    How fast is the level in the pot rising when the coffee in the cone is 5 in. deep?
        (b)    How fast is the level of coffee in the cone falling at that moment?

Difficulty Rating: 

(Thanks to Asuka for her help with this monster...)

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Solution: (a)    We are going to use the formula for volume of a cylinder. It may worry you that we don't know the height of the liquid in the pot (we only know that the cone height is 5...), but it turns out that we won't need to know it, as the radius of the liquid in the coffee pot is constant (unlike the cone) regardless of height, so dr/dt = 0: Decimal answers are certainly okay, as long as they're accurate to the thousandths place.

(b)    The big insight in this problem is setting up a set of similar triangles (like in Problem 9), and putting r in terms of h, as shown in the below diagram.

Now you can find dh/dt.

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