18. rectangles beneath a parabola a rectangle is constructed with its base on the x - axis and two of its…

Explanation:

Step1: Set up the area function

Let the (x) - coordinate of the right - hand vertex of the rectangle on the parabola be (x). The base of the rectangle is (b = 2x) (since the rectangle is symmetric about the (y) - axis) and the height (h=y = 48 - x^{2}). The area function (A(x)) of the rectangle is (A(x)=b\times h=(2x)(48 - x^{2})=96x-2x^{3}), where (x>0).

Step2: Find the derivative of the area function

Differentiate (A(x)) with respect to (x) using the power rule ((x^{n})^\prime=nx^{n - 1}). (A^\prime(x)=\frac{d}{dx}(96x-2x^{3})=96 - 6x^{2}).

Step3: Find the critical points

Set (A^\prime(x) = 0), so (96 - 6x^{2}=0). Rearrange the equation: (6x^{2}=96), then (x^{2}=16). Since (x>0), we get (x = 4).

Step4: Check the second - derivative

Differentiate (A^\prime(x)) to get the second - derivative (A^{\prime\prime}(x)=\frac{d}{dx}(96 - 6x^{2})=-12x). When (x = 4), (A^{\prime\prime}(4)=-12\times4=-48<0). So the function (A(x)) has a maximum at (x = 4).

Step5: Find the dimensions and area

When (x = 4), the base (b = 2x=8). The height (h=48 - x^{2}=48-16 = 32). The area (A=(2x)(48 - x^{2})), substituting (x = 4) gives (A=8\times32 = 256).

Answer:

The base of the rectangle is (8), the height is (32), and the maximum area is (256).