Answer:
Step-by-step explanation:
Given that a rectangle is constructed with its base on the x axis and two of its vertices on the parabola
[tex]y=100-x^2[/tex]
This parabola has vertex at (0,100) and symmetrical about y axis.
Any general point above x axis can be written as (a,b) (-a,b) since symmetrical about yaxis.
Hence coordinates of any rectangle are
[tex](a,0) (-a,0), (a, 100-a^2), (-a, 100-a^2)[/tex]
Length of rectangle = 2a and width = [tex]100-a^2[/tex]
Area of rectangle = lw = [tex]2a(100-a^2)=200a-400a^3[/tex]
To find max area, use derivative test.
[tex]A' = 200-800a^2\\A"=-1600a<0[/tex]
Hence maxima when first derivative =0
i.e. when a =2
Thus we find dimensions of the rectangle are l =4 and w = 96
Maximum area = [tex]4(96) = 384[/tex]
