Question

Find the maximum area of a rectangle inscribed beneath the parabola y = -x^2 + 6 where one side of the rectangle lies on the x-axis.

A) 9 square units
B) 12 square units
C) 16 square units
D) 18 square units

Answer 1

To find the maximum area of a rectangle inscribed beneath the parabola y = -x^2 + 6, we need to consider the dimensions of the rectangle. The maximum area will occur when the rectangle is a square, so we need to find the x-coordinate of the vertex of the parabola. The maximum area of the rectangle is 36 square units.

Step-by-step explanation:

To find the maximum area of a rectangle inscribed beneath the parabola y = -x^2 + 6, we need to consider the dimensions of the rectangle. Since one side of the rectangle lies on the x-axis, the base of the rectangle will be determined by the x-coordinates where the parabola intersects the x-axis. The maximum area will occur when the rectangle is a square, so we need to find the x-coordinate of the vertex of the parabola. The x-coordinate of the vertex can be found using the formula x = -b/(2a) for a parabola in the form y = ax^2 + bx + c.

In this case, the parabola is y = -x^2 + 6, so a = -1, b = 0, and c = 6. Substituting these values into the formula, we get x = -0/(2(-1)) = 0. Therefore, the maximum area of the rectangle is 6^2 = 36 square units.