Respuesta :
Answer:
Option B.
Step-by-step explanation:
The given curve is
[tex]y=27-x^2[/tex]
We need to find the area of the largest rectangle with lower base on the x-axis and upper vertices on the curve [tex]y=27-x^2[/tex].
Let the vertex in quadrant I be (x,y), then the vertex in quadrant II is (-x,y) .
Length of the rectangle = 2x
Width of the rectangle = y
Area of a rectangle is
[tex]Area=Length\times width[/tex]
[tex]Area=2x\times y[/tex]
Substitute the value of y from the given equation.
[tex]Area=2x(27-x^2)[/tex]
[tex]A=54x-2x^3[/tex] .... (1)
Differentiate with respect to x.
[tex]\frac{dA}{dx}=54-6x^2[/tex]
Equate [tex]\frac{dA}{dx}=0[/tex], to find the critical points.
[tex]0=54-6x^2[/tex]
[tex]6x^2=54[/tex]
Divide both sides by 6.
[tex]x^2=9[/tex]
[tex]x=\pm 3[/tex]
The value of x can not be negative because side length can not be negative.
Substitute x=3 in equation (1).
[tex]A=54(3)-2(3)^3[/tex]
[tex]A=162-54[/tex]
[tex]A=108[/tex]
The area of the largest rectangle is 108 square units.
Therefore, the correct option is B.
