Coordinate geometry doubt

Question

OABC is a rectangle where A lies on x-axis, C lies on y-axis and B lies on the curve x 2 3 + y 2 3 = a 2 3 , O is the origin.

Find the coordinates of B so that the area of OABC is maximum.

Answer 1

The origin and point B are opposite vertices of the rectangle, so consider the rectangle that lies in the 1st quadrant.

The coordinates of B can be specified as [x, (a^2/3 - x^2/3)^3/2].

The area of the rectangle R(x) is the product of those coordinates:

R(x) = x[ (a^2/3 - x^2/3)^3/2]

Determine R'(x) using the product rule:

R'(x) = x (3/2)[(a^2/3 - x^2/3)]^1/2(-2/3)x^-1/3 + [(a^2/3 - x^2/3)]^3/2

= - x^2/3[(a^2/3 - x^2/3)]^1/2 + [(a^2/3 - x^2/3)]^3/2

= [a^2/3 - x^2/3]^1/2 • [ -x^2/3 + a^2/3 - x^2/3]

The critical numbers occur when R'(x) equals zero (the first factor results in x = a, so y = 0).

The 2nd factor results in:

2x^2/3 = a^2/3

x^2/3 = (1/2)a^2/3

x = (1/2)^3/2 a = √(1/8) a = (√2/4)a ≈ 0.354a

Plugging one of those values into the original equation and solving for y results in y = (√2/4)a.

You can take the time to apply the 1st derivative test to show that the point [(√2/4)a, (√2/4)a] results in a maximum area.

desmos.com/calculator/jtqkhlwwv5

Use the slider on point A to move it to where first derivative equals zero (max area).