A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola . What are the width and height of such a rectangle with the greatest possible area?
You need to provide the formula for the parabola
To find the rectangle with the greatest possible area, we need to maximize the area of the rectangle. Let's break down the problem into steps:
Step 1: Understand the problem
A rectangle is inscribed in a parabola with its base on the x-axis. We need to find the dimensions of the rectangle (width and height) that maximize its area.
Step 2: Identify what we know
- The rectangle's base is on the x-axis, which means it has a length (width) equal to the x-coordinate of the upper corners of the rectangle.
- The upper corners of the rectangle lie on the parabola, which means their y-coordinates are determined by the equation of the parabola.
Step 3: Define the problem mathematically
We need to find the width and height of the rectangle that maximize its area. Let's denote the width as w and the height as h.
Step 4: Determine the equation of the parabola
The equation of a general parabola is of the form y = ax^2 + bx + c. Since the parabola intersects the x-axis at its base, the y-coordinate of the upper corners of the rectangle is given by the equation of the parabola.
Step 5: Express the area of the rectangle in terms of w and h
The area of a rectangle is given by A = w * h.
Step 6: Maximize the area
To maximize the area A, we need to find the values of w and h that result in the largest possible value of A.
To do this, we can express A (area) as a function of w only, by substituting the height h in terms of w based on the equation of the parabola. Then we can take the derivative of A with respect to w, set it equal to zero, and solve for w. This will give us the width (w) that maximizes the area.
Once we have the value of w, we can substitute it back into the equation of the parabola to find the corresponding height (h).
Step 7: Solve for w and h
By solving the equation obtained in step 6, we can find the width (w) that maximizes the area. Then, substitute this value into the equation of the parabola to find the corresponding height (h).
Step 8: Answer the original question
Once we have the values of w and h, we can answer the original question by stating the width and height of the rectangle with the greatest possible area.
Note: The specific equation of the parabola is not provided in the question, so you would need to know or be given the equation of the parabola to solve this problem.