A rectangle is constructed with its base on the x-axis and two of its vertices o

Question

A rectangle is constructed with its base on the x-axis and two of its vertices on the parabola y = 16 - x2. What are the dimensions of the rectangle with the maximum area? What is the area? The shorter dimension of the rectangle is and the longer dimension is . (Round to two decimal places as needed.)

Explanation / Answer

Suppose two of the vertices lie on the x axis be (x,0) and (-x,0). The other two verices => (x, 16-x^2) and (-x, 16-x^2). Hence length of rectangle = 2x units breadth= 16-x^2 units Then A = lenth * breadth A = 2x(16-x^2) = 32x - 2x^3 dA/dx = 32- 6x^2 = 0 6x^2 = 32 ==> x=sqrt(16/3) y = 16-x^2 = 32/3 dimesions that give the largest area are =sqrt(16/3) * 32/3

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