Formula for a bounded shape (rectangle)

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June 15th 2012, 08:50 AM
daigo

Formula for a bounded shape (rectangle)

Quote:

A rectangle is bounded by the x-axis and the semicircle y = sqrt(25 - x^2)
Express the area of the rectangle as a function of x and find the domain of the function

So this is how I approached the problem and got stuck:

http://i.imgur.com/UBVrU.png

But apparently this is how you're supposed to visualize it:

http://i.imgur.com/7HWIm.png

What I don't understand is how the side of the rectangle can be represented as the same thing as the formula for the semicircle. I thought they had to be different variables because they're two different things. How do you know you can use the same variables for both the side of the rectangle and the equation for the semicircle? And why is the other side of the rectangle 2x and not just x?

For the domain, it's apparently 0 ≤ x ≤ 5
But for the formula of the rectangle given in the correct solution, how can an area be 0? I would've thought you can't include 0 in the domain since then there wouldn't be any rectangle.
June 15th 2012, 09:07 AM
Reckoner

Re: Formula for a bounded shape (rectangle)

Quote:

Originally Posted by daigo

What I don't understand is how the side of the rectangle can be represented as the same thing as the formula for the semicircle. I thought they had to be different variables because they're two different things.

Are they different things? The equation gives us the $y$ -coordinate of a point on the semicircle corresponding to a particular $x$ -coordinate. But at one particular $x$ -coordinate (which we can simply call $x$ ), the height of the rectangle and the height of the semicircle are equal. Therefore, for this particular $x,$ the height of the rectangle is $y,$ and $y$ is given by $y = \sqrt{25-x^2}.$

Quote:

Originally Posted by daigo

And why is the other side of the rectangle 2x and not just x?

$x$ is the distance from the $y$ -axis to one side of the rectangle. To get the distance from the left side to the right side, we add the two distances in between: $x + x = 2x.$
June 15th 2012, 09:12 AM
HallsofIvy

Re: Formula for a bounded shape (rectangle)

With the given coordinate system, so that the center of the circle is at (0, 0), the corner of the rectangle in the first quadrant is $(x, y)= (x, \sqrt{25- x^2})$ . You know that "you can use the same variables for both the side of the rectangle and the equation for the semicircle" because they are layed out on the same coordinate system. And (x, y) depend on the coordinate system, not the figures. And the side is "2x and not just x" because x, the coordinate, is measured from the origin of the coordinate system, the center of the circle.

Whether you consider a single line segment as a "degenerate rectangle" or not is a matter of style. If you are arguing this with friend, go ahead and take whatever side you want. If your teacher asserts it is a rectangle with 0 area, and you want to pass the course, agree with your teacher!
June 15th 2012, 04:57 PM
Wilmer

Re: Formula for a bounded shape (rectangle)

Why don't you erase the darn semi-circle and see what happens!