Question

A gardener has 40 feet of fencing with which to enclose a garden adjacent to a long existing wall. The gardener will use the wall for one side and the available fencing for the remaining three sides.

If the sides perpendicular to the wall have length x feet, which of the following (A, B, C, or D) represents the area A of the garden?

A. A(x) = –2x^2 + 20x
B. A(x) = –2x^2 + 40x
C. A(x) = 2x^2 – 40x
D. A(x) = x^2 – 40x

I say it is B. A(x) = –2x^2 + 40x, any thoughts?

The area function is a quadratic function and so its graph is a parabola.

Does the parabola open up or down? I say down, am I correct?

Find the vertex of the quadratic function:

(-b/2a, f (-b/2a))

(0+20)/2 =10

-2x^2+40x = 200

Vertex = (10,200)

Is my vertex correct?

Use the work in the previous parts to help determine the dimensions of the garden which yield the maximum area, and state the maximum area. (Fill in the blanks below. Include the units of measurement.)

The maximum area is 2000 sqft. Is this accurate?

-2x^+40x
After differentiating and equating to 0
= -4x + 40=0
Putting this value in area function..
Or.
in the perimeter function.
4x=40

2x+y=40
Put x=10
y=40-20
y=20
x=10

200*10 = 2000 sqft

When the sides perpendicular to the wall have length x = 10ft

and the side parallel to the wall has length 200ft

Are these calculations correct based on my original answer of
B. A(x) = –2x^2 + 40x

Any feedback is very much appreciated! Thanks!

Answer 1

all correct until you maximize
vertex at (10,200) is right.
That is the top of the parabola, maximum
A = -2x^2+40 x
A = -2(100) + 40(10)
A = 200
or using calculus
dA/dx = 0 = -4x + 40
x = 10
Then the same again
Clearly the side opposite the wall can not be 200 feet if you only have 40 feet of fence

Answer 2

So then everything is correct? What do I need to change?

When the sides perpendicular to the wall have length x = 10ft?

and the side parallel to the wall has length 200ft Or should this be 20ft?

The maximum area is 2000 sqft?

Thanks!

Answer 3

Your vertex of the A(x) area function is (10, 200), so the maximum area is 200 square feet.

x is the length of the sides perpendicular to the wall, and 200/x is the length of the side parallel to the wall.

x=10, so the length of the sides perpendicular to the wall is 10ft, and the length of the side parallel to the wall is 20ft