Final answer:
To find the area of the rectangle, we can express it as a function of x and simplify the expression. The domain of the function is [0, 6], and the graph of the function is a downward-opening parabola. The value of x where the area is largest is 18.
Step-by-step explanation:
(a) To find the area of the rectangle, we need to find the length and width of the rectangle. The length is the distance between the origin and the corner in quadrant I, which can be found by solving the equation y = 36 - x^2 for x. The width is the distance between the corner on the positive y-axis and the origin, which is y. Therefore, the area A of the rectangle can be expressed as:
A(x) = x * y
To simplify the expression, we substitute the value of y from the equation y = 36 - x^2:
A(x) = x * (36 - x^2)
(b) The domain of A(x) is the set of all x-values that make sense in the context of the problem. Since x represents a distance, it cannot be negative and should be less than or equal to 6 (the maximum x-value that satisfies the quadratic equation). Therefore, the domain of A(x) is [0, 6].
(c) To graph A(x), we plot the x-values from the domain on the x-axis and the corresponding A(x) values on the y-axis. The graph will be a downward-opening parabola since A(x) = -x^3 + 36x. To find the value of x where A(x) is largest, we find the vertex of the parabola. The x-coordinate of the vertex is given by x = -b/2a, where a = -1 and b = 36. Plugging in these values, we get x = -36/(2*(-1)) = 18. Therefore, the value of x where A(x) is largest is 18.