Pre-calculus, last question before the EXAM, help please?

Question

I understand this question partially, but i'm not sure where my teacher got A=x^2+xy

Here's the question:

A farmer wishes to build two enclosures using fencing. One enclosure must be a square, and the second must be a rectangle with the length of one side equal to the side length of
the square. The two enclosure are not connected. 1000 m of fencing is available for the project

1a. Let x represent the side length of the square and A(x) the total area of both enclosures as a function of x. Find a formula fot A(x).

1b. Find the dimensions of the square and rectangle which enclose the largest area possible. (The solution may appear obvious to you, but prove it using A(x) from part (a).)

Thank you!

Answer 1

the sides of the square are x, its area is x², its perimeter is 4x. the length and width of the rectangle are x and y, its area is xy, its perimeter is 2x + 2y.

total perimeter is 6x + 2y = 1000
total area is x² + xy = A(x), and we want that maximized. solve the perimeter equation for y:
2y = 1000 - 6x
y = 500 - 3x

and plug that into the area equation:
A(x) = x² + x(500 - 3x)
A(x) = x² + 500x - 3x²
A(x) = -2x² + 500x
A(x) = 2x(-x + 250)

you know from algebra 1 that that's a parabola, open down, with x intercepts at 0 and 250. you also know the vertex is halfway between those intercepts at x = 125.

when x = 125,
6(125) + 2y = 1000
750 + 2y = 1000
2y = 250
y = 125

Answer 2

A = x^2 + xy
is a formula for the area. The x^2 is the area of the square and xy is the area of the rectangle where y is the other side.
But you need y to proceed. That's where the 1000 m comes in; its the total perimeter.

1000 = 4x + 2(x+y)
1000 = 6x + 2y Solve this for y and we can plug it into the A formula to have a function of just x
500 = 3x + y
y = 500 - 3x

A(x) = x^2 + x(500 - 3x)
A(x) = 500x - 2x^2

for part b. you need the derivative to be zero, so find the derivative and set equal to zero.

500 - 4x = 0
4x = 500
x = 125
So the largest area is when x = 125
What is supposed to be obvious is y is also 125 so both enclosures are squares.

Answer 3

Let the two sides of the rectangulat be x an y, where x in the side of the square

now area of rectangle=x*y
and area of square=x*x

therefore total area=x*x+x*y

now 1000=4*x+2(x+y)

500=3*x+y or y=500-3x

therefore A(x)=x*x+xy
=x^2+x(500-3x)

Answer 4

If x is side of the square, its perimeter is 4x. The 2 sides of rectangle, x each, will add to 2x. If 2y is addition of other 2 sides then,by data, 1000=4x+2x+2y . then y={1000-6x}/2
then total area of 2 figures is=x^2+xy=x^2+x(1000-6x)/2
then for Amax dA/dx=0 or 500-4x=0.0 Hence x=125 and y=125 also. Both areas are squares