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Bavmorda's Hat - DMS Discussion

Bavmorda's Hat - DMS Discussion
AdminZIML - 2017年11月8日 Wednesday 15:48

From the week of October 30 to November 3 we chose Wednesday's problem (that you can find here):

Bavmorda, the witch, is out shopping for fabric. She needs to make a new hat for herself, as her old one got caught in a cauldron explosion. To build her hat she will need to create a cone with no base and will add to it a ring of fabric to make the flap. She needs the cone to have a diameter of $5$ inches to fit her head comfortably, and she wants the ring around the hat to be $4$ inches wide. If the hat has to be $10$ inches tall, how many square inches of fabric should Bavmorda buy?

Recall that the formula for surface area of a cone with no base is $\pi rs$, where $r$ is the radius of the base and $s$ is the length from the edge of the circle to the tip of the cone. Use $\pi=3.14$ and round your answer to the nearest tenth if necessary.

25% of the students who tried the problem got it right on their first attempt.

The problem wants us to find how much fabric should Bavmorda buy, that is, we want to find the surface area of the cone (with no base) and the surface area of the ring that goes around the hat.

We are provided with a formula for the surface area of the cone portion of the hat, however, we cannot use it right away as we do not know the length from the edge of the circle to the tip of the cone. A common mistake was to use $s = 10$ inches in the formula. To figure out missing information it is useful to make a diagram. In this problem a diagram of the side view of the hat helps:

We can see then it would be possible to find the missing information by using the Pythagorean Theorem. Afterwards we just need to be extremely careful plugging in the values of $r$ and $s$ in the formula. What else do we need to do to finish the problem?

Share your thoughts and questions below!