So, I haven't posted in a while...but now I am stummped! Only question I can't get today!

A team of students seeks to make a flag representing their commitment to the earth. They will be using a green triangle inscribed in a yellow semi-circle. Find the maximum area of the triangle inscribed in a semi-circle of radius r.

So I know you somewhere have to use 1/2pir^2...but then what?

THANKS!

Sounds like a common sense kind of problem

The largest area is obtained with a largest possible base and a largest possible height.
Clearly the diameter is as large as you could make a base, and the height cannot be any larger than the radius at from the centre of the semi-circle
so the largest area would be 1/2(2r)(r) = r^2

You could try putting that if the base is = r and the height is r^2 and the triangle is isosceles, you will get the largest area. I think!

To find the maximum area of the triangle inscribed in a semi-circle of radius r, we can follow these steps:

Step 1: Draw a diagram: Draw a semi-circle with radius "r" and a triangle inscribed inside it. The triangle should touch the semi-circle at three points - the midpoint on the semi-circle's diameter, and the two endpoints of the diameter.

Step 2: Identify the dimensions: Let the base of the triangle be "b" and the height of the triangle be "h". We want to find the maximum possible area (A) of the triangle.

Step 3: Use the formula for the area of a triangle: The area of a triangle is given by the formula A = (1/2) * base * height.

Step 4: Relate the dimensions of the triangle: In the given figure, the base of the triangle is the diameter of the semi-circle, which has a length of 2r. The height of the triangle is equal to the radius of the semi-circle, which is r.

Step 5: Substitute the dimensions in the formula: Substitute the values of the base and height into the area formula: A = (1/2) * 2r * r = r^2.

Step 6: Determine the maximum area: To find the maximum area of the triangle, we need to find the maximum value of r^2. Since r^2 will always be positive, there is no maximum value for r^2. Therefore, the maximum area of the triangle inscribed in a semi-circle is infinity (assuming there are no other constraints or limitations).

In summary, the maximum area of the triangle inscribed in a semi-circle of radius r is infinity, as there is no limit to the value of r^2.