modelling with functions

[watchthesky30]

one of length x and the other of length 10-x. Each piece is bent into the shape of a square.
a) Find a function that models the total area enclosed by the two squares.
I already did this. & I got the function A(x)=-(x+5)²+25
but i asked somebody and he got a totally different answer.
his answer:
perimeter of square 1 is x, so the side is x/4. The area is x² / 16
perimeter of square 2 is (10 - x) so the side is (10 - x) / 4 and the area is (100 - 20x - x²) / 16
Add the two equations to get
A(x) = x² / 16 + (100 - 20x + x²) / 16
= (100 - 20x + 2x²) / 16
= (50 - 10x + x²) / 8

is he wrong? am i wrong? or are we both wrong?

b) Find the value of x that minimizes the total area of two squares.
how do i solve and get the answer for this question? Please help .

 

[BigGlenntheHeavy]

Hint: The domain of the two lengths is [0,10].

 

[DrMike]

one of length x and the other of length 10-x. Each piece is bent into the shape of a square.
a) Find a function that models the total area enclosed by the two squares.
I already did this. & I got the function A(x)=-(x+5)²+25
but i asked somebody and he got a totally different answer.
his answer:
perimeter of square 1 is x, so the side is x/4. The area is x² / 16
perimeter of square 2 is (10 - x) so the side is (10 - x) / 4 and the area is (100 - 20x - x²) / 16
Add the two equations to get
A(x) = x² / 16 + (100 - 20x + x²) / 16
= (100 - 20x + 2x²) / 16
= (50 - 10x + x²) / 8

is he wrong? am i wrong? or are we both wrong?

b) Find the value of x that minimizes the total area of two squares.
how do i solve and get the answer for this question? Please help .

Your friend is correct.

To solve part (b), you need to find the minimum of the parabola (50-10x+x^2)/8

 

[watchthesky30]

thank you . And sorry, but i was also wondering if i could do it another way. Like this:

If total length is "x", each side of the square is x/4
and the area is (x/4)^2
----------------------
If total length is "10-x", each side of the square is (10-x)/4
and the area is [(10-x)/4]^2
----------------------------
a) Find a function that models the total area enclosed by the two squares.
Total Area = (x/4)^2 + [(10-x)/4]^2
TA = (x^2/16) + [(100 -20x + x^2)/16]
TA = (1/8)x^2 -(5/4)x + (25/4)
---------------------------------------------

b) Find the value of x that minimizes the total area of two squares.
Min occurs when x = -b/2a = (5/4)/[2(1/8)] = (5/4)/(1/4) = 5

or is this way wrong