Maximum Area

[nycmathguy]

Section 2.1
Question 73

For this question, I seek a hint for parts a & b.

 

[MathLover1]

a hint for parts a : the length of corrals is 2x, the width is y
fencing or perimeter=200ft

& b: to determine the maximum first find x and y

 

[nycmathguy]

a hint for parts a : the length of corrals is 2x, the width is y
fencing or perimeter=200ft

& b: to determine the maximum first find x and y

Thanks for the hints. Will work on this and show my work.

 

[nycmathguy]

a hint for parts a : the length of corrals is 2x, the width is y
fencing or perimeter=200ft

& b: to determine the maximum first find x and y

Part (a)

P = 200

200 = 2(2x) + 2y

200 = 4x + 2y

200 = 2(2x + y)

200/2 = 2x + y

100 = 2x + y

(100 - 2x) = y

A(x) = length •width

A(x) = 2x (100 - 2x)

A(x) = 200x - 4x^2

Tell me if this right before moving on to part (b).

 

[MathLover1]

correct

 

[nycmathguy]

correct

I will do part (b) tomorrow.

 

[nycmathguy]

correct

A(x) = 200x - 4x^2

Let A(x) = 0

0 = 200x - 4x^2

I found x to be 0 and 50. Since we are dealing with measurements in terms of feet, I rejected x = 0.

Now that I found x to be 50, where do I plug my x-value to find y?

 

[MathLover1]

in part a
200 = 2(2x) + 2y

 

[nycmathguy]

in part a
200 = 2(2x) + 2y

Let x = 50

200 = 2(2x) + 2y

200 = 2(2•50) + 2y

200 = 2(100) + 2y

200 = 200 + 2y

200 - 200 = 2y

0 = 2y

0/2 = y

0 = y

What does x = 50 & y = 0 represent here?
Has part (b) been answered?

 

[MathLover1]

I gave you bad hint, a hint for part a : the perimeter=200ft includes the fence in the middle, the shared fence

so you have perimeter

Let x represent the dimension perpendicular to the shared fence. (these 2x)
Then the dimension parallel to the shared fence will be: (3y)
3y = 200-2x
y = (200-2x)/3
y = (2/3)(100-x)

The area will be the product of these dimensions, so will be:

area = x*y
area = x(2/3)(100-x)
area = (2/3)(100x -x^2)
area = (-2/3)(x^2 -100x)

To complete the square, we need to add the square of half the x-coefficient inside parentheses, and the opposite of that quantity outside parentheses.

area = (-2/3)(x^2 -100x+50^2)-(-2/3)50^2. . . . complete the square

area = (-2/3)(x -50)^2 +5000/3

The vertex of this parabolic curve is at x=50, so the dimensions of the maximum area is at x = 50
then substitute it in
y = (200 -2*x)/3
y = (200 -2*50)/3
y=100/3
y= 33 &1/3 feet

 

[nycmathguy]

I gave you bad hint, a hint for part a : the perimeter=200ft includes the fence in the middle, the shared fence

so you have perimeter

Let x represent the dimension perpendicular to the shared fence. (these 2x)
Then the dimension parallel to the shared fence will be: (3y)
3y = 200-2x
y = (200-2x)/3
y = (2/3)(100-x)

The area will be the product of these dimensions, so will be:

area = x*y
area = x(2/3)(100-x)
area = (2/3)(100x -x^2)
area = (-2/3)(x^2 -100x)

To complete the square, we need to add the square of half the x-coefficient inside parentheses, and the opposite of that quantity outside parentheses.

area = (-2/3)(x^2 -100x+50^2)-(-2/3)50^2. . . . complete the square

area = (-2/3)(x -50)^2 +5000/3

The vertex of this parabolic curve is at x=50, so the dimensions of the maximum area is at x = 50
then substitute it in
y = (200 -2*x)/3
y = (200 -2*50)/3
y=100/3
y= 33 &1/3 feet

Great reply. Don't worry about giving me the wrong hint. It happens to everyone. More math later from the beach.

 

[MathLover1]

I just don't know why question was to express the area in terms of x when there is much faster and easier way to get same answer

given perimeter 200ft
half of it is 2 times the length 2x=>means 2*2x=100=>the length 2x=50ft
half of it is 3 times the width 3y=>means 3y=100=>the width y=100/3 =33&1/3ft=33.33ft

then area=2x*y=50ft*33.33ft=1666.5ft^2

 

[nycmathguy]

I just don't know why question was to express the area in terms of x when there is much faster and easier way to get same answer

given perimeter 200ft
half of it is 2 times the length 2x=>means 2*2x=100=>the length 2x=50ft
half of it is 3 times the width 3y=>means 3y=100=>the width y=100/3 =33&1/3ft=33.33ft

then area=2x*y=50ft*33.33ft=1666.5ft^2

I like the shortcut you provided.