A venetian blind (made of horizontal strips) hangs over a circular window that is 8' in diameter. The window is revealed as the blinds are raised, and they can be raised to a height of 6.5 feet above the window's lower-most point. The window's uncovered area increases as the blinds' height increases.

**Problem 1** (from class): Use a Riemann sum to define a function that approximates the uncovered area of window, expressed as a function of how high the blinds are above the window's bottom-most point.

*Solution from class:*

*x* = height of blinds above window's bottom, in feet

0 ≤ *x* ≤ 6.5

m = 200

h=.05

*w*(*x*) =

*(Note: type ctrl-R to get "", the square root sign)
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*A*(*x*) =

*
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**Problem 2** (from class): Construct a function that approximates the rate of change of uncovered area with respect to its height, where the function is expressed as a function of the blinds' height.

*Solution from class:*

*A*(*x*) =

*
*

*r*(*x*) =

**Problem 3** (for you, **now**)**:**

Why do the graphs of *r*(*x*) and *w*(*x*) appear to coincide?

(I'm looking for a *conceptual* explanationÑan explanation that makes clear what is going on.)*
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