Question

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

Answer 1

108 units squared

Explanation:

Length is 2x , width is y

area = 2x * y

sub for y

area = 2x(27 - x^2)

area = 54x - 2x^3

1st derivative: 54 - 6x^2

54 - 6x^2 = 0

6(9 - x^2) = 0

6(3 - x)(3 + x) = 0

so, x = 3

solve for y: 27 - 3^2 = 18

area = (3*2)* 18 = 108 units squared

Answer 2

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