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### Related Rates: Circle Inscribed in an Expanding Square

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Date: 03/01/99 at 23:07:26
From: Brian Pollock
Subject: Calculus - Related Rates

I am having a hard time getting anywhere on this problem.

A circle is inscribed in a square with that circle tangent to each side
of the square. The circumference of the circle is increasing at a
constant rate of 6 inches per second. As the circle expands, the square
expands to maintain the condition of tangency.

A. Find the rate at which the PERIMETER of the square is increasing.
Indicate units of measure.

B. At the instant when the area of the circle is 25(pi) square inches,
find the rate of increase in the AREA ENCLOSED BETWEEN the circle
and square. Indicate units of measure.
```

```
Date: 03/02/99 at 08:22:56
From: Doctor Jerry
Subject: Re: Calculus - Related Rates

Well, the perimeter of the square is 4s, where s is one side. The side
of the square must be the diameter of the circle, which is 2r, where r
is the radius of the circle. So, P = 4s = 4(2r) = 8r.

We also know that C=2pi*r and dC/dt = 6 in/s.  So, dC/dt = 2pi*dr/dt

and

dr/dt = (dC/dt)/(2pi) and so know we know dr/dt.

dp/dt = 8dr/dt, right?  So know we know dp/dt.

AB = area between circle and square is s^2-pi*r^2 = 4r^2-pi*r^2

So, d(AB)/dt = (4-pi)2*r*dr/dt.

Figure out what r would be when the area of the circle is 25pi sq.in.
and substitute into the above equation.

- Doctor Jerry, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Calculus

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