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vovikov84 [41]
2 years ago
12

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
Mathematics
1 answer:
LUCKY_DIMON [66]2 years ago
7 0

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.

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