The dimensions that give the largest area for the rectangle are:
- Height: 2.79 in
- Width: 2.47 in
The largest area is A = 12.97 in².
Modeling:
Since the base of the rectangle is on the x-axis and its other two vertices are above the x-axis, lying on the parabola y = 7 - x², we can represent the height of the rectangle by the expression 7 - x². Let the width of the rectangle be represented by x.
The area of the rectangle is given by A = (height)(width) = (7 - x²)x.
Optimizing:
To find the dimensions that give the largest area for the rectangle, we need to maximize the function A = (7 - x²)x.
We can maximize the function using the following steps:
Find the derivative: A'(x) = -7x + 2x³
Set the derivative equal to zero and solve: -7x + 2x³ = 0. The solutions are x = 0 and x = \frac{3.5 \sqrt{2}}{2}.
Evaluate the derivative at each critical point and the endpoints of the domain: The function's domain is 0 ≤ x ≤ 3.5. Evaluating the derivative, we get:
A'(0) = 0
A'(3.5) = 18 > 0
Analyze the intervals:
For
, so the function is increasing.
For
, so the function is decreasing.
Conclusion:
Based on the analysis of the intervals, we can conclude that the maximum value of the function occurs at x = \frac{3.5 \sqrt{2}}{2}. Therefore, the dimensions that give the largest area for the rectangle are:
The largest area is
≈ 12.97 in².