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.
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
and the circumference is given by
Solve the circumference equation for r and substitute the value of r into the area expression to get
A and C are both functions of time so it is more explicit to write
If you now differentiate both sides with respect to t you get
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?