Question

What is the area of the largest rectangle with lower base on the x-axis and upper vertices on the curve y = 27 − x^2? show your work.
answer choice

324

108

18

3

Answer 1

Answer:

Option B.

Step-by-step explanation:

The given curve is

$y=27-x^2$

We need to find the area of the largest rectangle with lower base on the x-axis and upper vertices on the curve $y=27-x^2$.

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

$Area=Length\times width$

$Area=2x\times y$

Substitute the value of y from the given equation.

$Area=2x(27-x^2)$

$A=54x-2x^3$      .... (1)

Differentiate with respect to x.

$\frac{dA}{dx}=54-6x^2$

Equate $\frac{dA}{dx}=0$, to find the critical points.

$0=54-6x^2$

$6x^2=54$

Divide both sides by 6.

$x^2=9$

$x=\pm 3$

The value of x can not be negative because side length can not be negative.

Substitute x=3 in equation (1).

$A=54(3)-2(3)^3$

$A=162-54$

$A=108$

The area of the largest rectangle is 108 square units.

Therefore, the correct option is B.