Quandaries and Queries



My name is John and I am in 12th grade in San Diego in a Calc class for the summer. There was a question the teacher asked and I was wondering if you could help. It is a related rate problem dealing with a spiral. Here it goes.....A mathematics professor is knitting a sweater. The main part of the sweater is knit in a large spiral, ending up with a diameter of 30 inches. She knits at a constant rate of 6/7 square inches per minute.

1. At what rate is the circumference of the circle increasing when the diameter is 2 inches?

2. How long will it take her to finish this piece of the sweater?

If you can help I would appreciate it.




Hi John,

I'll try to get you started. You are told the rate at which the area of the area is increasing (6/7 square inches per minute) and you need to determine the rate at which the circumference is increasing. Thus you need a relationship between the area and the circumference if a circular disk.

The area of a disk is given by

A = r2

and the circumference is given by

C = 2 r

Solve the circumference equation for r and substitute the value of r into the area expression to get

A = 1/4 1/ C2

A and C are both functions of time so it is more explicit to write

A(t) = 1/4 1/ C(t)2

If you now differentiate both sides with respect to t you get

A'(t) = 1/2 1/ C(t) C'(t)

You now have an expression that contains the rate at which the circumference is increasing at any time, C'(t).

Can you complete the problem now?



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