Write down the Cartesian equation, in terms of r, of a circle centered at the origin with a fixed radius r. Hence show that the area of a rectangle of length 2x that is inscribed in a circle is 4x(r^2-x^2)^0.5.
Find the value of r if the maximum area of the rectangle is 50.
Anonymous
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>>URGENT DIFFERENTIATION QUESTION?
No, it's not. It's a command for somebody to do one of your homework problems.
You should already know the general equation for a circle centered at the origin: x^2 + y^2 = r^2.
For some point x, the related y coordinate is y = ±√(r^2 - x^2). We'll use this in a moment.
If you pick point (x,y) on the circle that's in the first quadrant, and then use that as the top right corner of an inscribed rectangle, then the length of the rectangle is 2x. The height is 2y. But this is the same as 2√(r^2 - x^2). So the area is length times width, or 4x√(r^2 - x^2).
To find the maximum area, take the derivative with respect to x and set it equal to 0. Get this x in terms of r. Plug this back into the expression for the area, then set it equal to 50, and solve for r.