Sure, I can help you with that algebra question.
To find the area of the rectangle, we need to multiply the lengths of its adjacent sides. Let's call the length of the side on the x-axis x and the length of the side on the y-axis y.
From the given information, we know that one corner of the rectangle is at the origin, so its length on the x-axis is x, and its length on the y-axis is y = 36 - x^2 (based on the equation of the graph).
Therefore, the area A of the rectangle is: A = x * y = x(36 - x^2).
To determine the domain of x, we need to consider the restrictions on the values of x. Since one side of the rectangle is on the positive x-axis, x must be greater than 0. Additionally, based on the equation of the graph, we know that y = 36 - x^2. For y to be defined, the expression 36 - x^2 must be greater than or equal to 0 (since y cannot be negative). Solving this inequality gives us x^2 <= 36, which simplifies to |x| <= 6. From this, we can conclude that the domain of x is -6 <= x <= 6.
To find the value of x that maximizes the area (A), we can use calculus. We can differentiate the area function with respect to x, set it equal to zero, and solve for x. However, since you specifically asked for an explanation of how to get the answer and not just the answer itself, I assume you want a non-calculus explanation.
In this case, we can analyze the behavior of the function A = x(36 - x^2) by considering a sign chart. We can determine the sign of A for different intervals of x.
- If x < -6, then both x and 36 - x^2 are negative. Therefore, A = x(36 - x^2) is positive.
- If -6 < x < 0, then x is negative, but 36 - x^2 is positive. Therefore, A = x(36 - x^2) is negative.
- If x = 0, then A = x(36 - x^2) is zero.
- If 0 < x < 6, then both x and 36 - x^2 are positive. Therefore, A = x(36 - x^2) is negative.
- If x > 6, then both x and 36 - x^2 are positive. Therefore, A = x(36 - x^2) is positive.
From this analysis, we can conclude that the maximum value of A occurs at the boundary points x = -6, 0, and 6. We can evaluate A at these points to determine the maximum.
- At x = -6, A = -6(36 - (-6)^2) = -6(36 - 36) = 0.
- At x = 0, A = 0(36 - 0^2) = 0.
- At x = 6, A = 6(36 - 6^2) = 6(36 - 36) = 0.
Therefore, the maximum value of A is 0, and it occurs when x = -6, 0, or 6.