What Is the Area of the Largest Rectangle Inscribed in a Semicircle?

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The area of the largest rectangle inscribed in a semicircle with radius r is calculated using the formula A = r^2. To derive this, one can express one side of the rectangle in terms of the other, establishing a relationship between height and width. By drawing the semicircle and the inscribed rectangle, the dimensions can be analyzed to find the maximum area. The critical point is that knowing one dimension allows for the calculation of the other. Understanding these relationships is essential for solving the problem effectively.
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What is the area of the largest rectangle that can be inscribes indiside a semicircle with the radius r?

answer: x = r / SQRT(2)
A = r^2
 
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Is there a question here?
 
How do you ge the answer?
 
How did YOU get the answer?
 
Well, if you know one side of a rectangle inscribed in a semicircle, can you figure out the other side?
 
What Hurkyl's getting at is:

Draw the semicircle. Draw a rectangle inside it, call one side h, call the other side w, call the radius r.

Now find a relationship that let's you eliminate either h or w. In other words, express h as a function of w, or w as a function of h.
 

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