What is the area of the largest rectangle with lower base on the x-axis and upper vertices on the curve y = 27 - x2?

The answer is 54 square units.
let the vertex in quadrant I be (x,y) 
then the vertex in quadratnt II is (-x,y) 
base of the rectangle = 2x 
height of the rectangle = y 
Area = xy 
= x(27 - x²) 
= -x³ + 27x 
d(area)/dx = 3x² - 27 = 0 for a maximum of area 
3x² = 3 x 3² = 27 
x² = 9 
x = ±3 
y = 27-9 = 18 
So, the largest area = 3 x 18 = 54 square units

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