The figure shows a semicircle with radius 1, horizontal diameter PQ, and tangent lines at P and Q. At what height above the diameter should the horizontal line be placed so as to minimize the shaded area?
I would post the figure but the website is not allowing it.
I've tried using the equation for a circle and got x^2 + y^2 = 1, and used that to solve for x, x=sqrt(1-y^2), to integrate but can't seem to get the answer.
What is the shaded area?
To find the height above the diameter that minimizes the shaded area, you need to determine the equation of the line that represents the placement of the horizontal line.
Let's assume that the horizontal line intersects the diameter at point A. To find the equation of this line, we need to consider the properties of tangents to a circle.
A tangent line to a circle at a given point is perpendicular to the radius of the circle at that point. Since the radius is perpendicular to the diameter, the line connecting the center of the circle to the point of tangency splits the diameter into two equal parts.
Let's label the point where the horizontal line intersects the diameter A. This means that the distance from P to A is equal to the distance from Q to A.
Now, consider the right triangle formed by the radius, the line connecting the center to point A, and the line segment parallel to the diameter that connects the point A to the semicircle.
Since the radius of the semicircle is 1, and the line segment connecting point A to the semicircle is the height that we want to find, let's call it h.
Using the Pythagorean theorem, we can express this relationship as:
(1 - h)^2 + x^2 = 1
(1 - h)^2 + (1 - x)^2 = 1
where x represents the distance from the diameter to the rightmost part of the horizontal line.
Substituting for x in the first equation using x = 1 - h, we get:
(1 - h)^2 + (1 - (1 - h))^2 = 1
(1 - h)^2 + (1 - 1 + h)^2 = 1
(1 - h)^2 + h^2 = 1
Expanding and simplifying the equation, we get:
1 - 2h + h^2 + h^2 = 1
2h^2 - 2h = 0
Factoring out the common factor of 2h, we have:
2h(h - 1) = 0
Solving for h, we have two possible solutions:
1. h = 0, which represents the horizontal line being placed right on the diameter.
2. h = 1, which represents the horizontal line being placed at the topmost point of the semicircle.
Since the question asks for the height that minimizes the shaded area, we can disregard the second solution (h = 1) since it would result in no shaded area.
Therefore, the height above the diameter where the horizontal line should be placed to minimize the shaded area is at h = 0, which means placing the horizontal line right on the diameter.