Find the largest rectangle (that is , the rectangle with largest area) that fits inside the graph of the parabola y=x^2 below the line y=a (a is an unspecified constant value), with the top side of the rectangle on the horizontal line y=a

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Answer 1

To find the largest rectangle that fits inside the graph of the parabola

below the line

, we need to determine the dimensions of the rectangle that maximize its area.Let's denote the x-coordinate of the vertex of the parabola as

. The equation of the parabola is

, so the vertex is at

. The line

intersects the parabola at two points,

and

, where

and

are the x-coordinates of the points of intersection.The rectangle that fits inside the parabola and below the line

has its top side on the line

and its bottom side on the parabola

. The width of the rectangle is

, and the height is

.The area of the rectangle is given by the product of its width and height, which is:

To maximize the area, we need to find the values of

and

that maximize this expression. Taking the derivative of

with respect to

and setting it equal to zero, we get:

Solving this equation for

, we find:

Substituting this value back into the equation for

, we get:

Therefore, the dimensions of the largest rectangle that fits inside the graph of the parabola

below the line

are:Width:

Height:

The maximum area of the rectangle is:

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