Final answer:
The area of the largest isosceles triangle inscribed in a circle with radius 6 inches is 36 square inches. The base is the diameter of the circle, and the height is the radius. The area can be written as a function of the height as A(h) = 6h.
Step-by-step explanation:
The area of the largest isosceles triangle that can be inscribed in a circle of radius 6 inches can be found by considering that the triangle's base will span across the diameter of the circle, and thus its length will be 12 inches. The height h of the triangle will reach from the midpoint of the diameter (the base of the triangle) to the point opposite on the circumference of the circle, which coincides with the radius when the triangle is isosceles and right-angled.
Using the formula for the area of a triangle, which is A = 0.5 × base × height, the base is 12 inches and the height is also equal to the radius r, which is 6 inches. Therefore, the area is A = 0.5 × 12 × 6, yielding an area of 36 square inches. The function of h, in this case, can be written as A(h) = 6h, and the maximum area occurs when h is maximal, coinciding with the radius of the circle.
Using the known value of π (pi), we can write the area as a function of the radius r: A(r) = πr². However, since the inscribed isosceles triangle is right-angled, its area is exactly half the area of the circle, leading us to the same result of 36 square inches.