Find the rectangle of largest area that can be inscribed in a semicircle of radius R, assuming that one side of the rectangle lies on the diameter of the semicircle.
NOTE: Let L denote the length of the side that lies on the diameter and H denote the height of the rectangle. Your answer will likely involve R.
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To find the rectangle of the largest area inscribed in a semicircle with radius R, we need to maximize the area of the rectangle. Let's break down the problem into steps to find the solution:
Step 1: Visualize the problem
Draw a semicircle with radius R and a rectangle inscribed in it. Notice that the rectangle has one side on the diameter, which we'll call L, and the other side perpendicular to the diameter, which we'll call H.
Step 2: Identify the constraints
We know that one side of the rectangle is on the diameter of the semicircle, so L is related to R. Additionally, the height of the rectangle, H, is constrained by the semicircle's radius R.
Step 3: Define the objective
We want to find the rectangle with the largest area, which would maximize the product of its sides (L * H).
Step 4: Formulate the problem mathematically
Since H is constrained by the radius R, we can express it as H = R - x, where x is the vertical distance from the top of the rectangle to the semicircle's curved edge.
Step 5: Express the area of the rectangle in terms of x
The area of the rectangle is given by A = L * H. Substituting H = R - x, we have A = L * (R - x).
Step 6: Maximize the area
To find the maximum area, we take the derivative of A with respect to x and set it equal to zero:
dA/dx = -L = 0
This implies that L = 0. Since the length of the sides of a rectangle cannot be zero, it means that L must be equal to the diameter of the semicircle, which is 2R.
Step 7: Calculate the maximum area
Substituting L = 2R into the area equation, we have A = 2R * (R - x).
To find the maximum area, we need to determine the optimal value of x. To do this, we look at the endpoints of the vertical distance x (from 0 to R) and evaluate the corresponding areas.
When x = 0, A = 2R * R = 2R^2.
When x = R, A = 2R * 0 = 0.
Therefore, the maximum area occurs when x = 0, resulting in a rectangle with sides 2R (along the diameter) and R (perpendicular to the diameter). The area of this rectangle is A = 2R * R = 2R^2.
Thus, the rectangle of largest area that can be inscribed in a semicircle of radius R, with one side on the diameter, has sides 2R and R, with an area of 2R^2.