Math geniuses I need your help on a pre-calc challenge problem!?

Question

I received a challenge question from my math teacher...and I think i have the right answer but I'm not sure.
Here's a link to the picture:
http://smg.photobucket.com/albums/v321/sugaswt17/?action=view¤t=math.jpg

We got a hint that on the side of 8, we can make a third variable, z like this:
http://smg.photobucket.com/albums/v321/sugaswt17/?action=view¤t=mathii.jpg

Find the area of the rectangle as a function of its length. I used y and got A=y*((8z-z²)/(√z²-y²)).

Not sure if this is right at all =( Any help would be greatly appreciated.

Answer 1

I think you need to get a formula for the area of the rectangle in terms of only one variable. I take the length to be x. So let's find the area in terms of x.

Use the suggestion to divide the side of length 8 into z and
8 - z. Then by similar triangles we have:

x/(8 - z) = 10/8 = 5/4
4x = 5(8 - z) = 40 - 5z
5z = 40 - 4x = 4(10 - x)
z = 4(10 - x)/5

By similar triangles we also have:

y/z = 6/10 = 3/5
5y = 3z
z = 5y/3

Now we have two formulas for z. Set them equal.

z = 4(10 - x)/5 = 5y/3
12(10 - x) = 25y
y = 12(10 - x)/25

Now we can calculate the area of the rectangle in terms of x alone.

Area = xy = x[12(10 - x)/25] = 12x(10 - x)/25

Answer 2

You need an answer in terms of either x or y alone (depending on which one you choose to call the length).

Note that since the side of length 10 and the side marked x are parallel, the angles at the top of the big triangle and the one with x and 8-z as two of its sides are the same. Call this angle θ. We know sin θ = 6/10 = 3/5 and cos θ = 8/10 = 4/5 from the big triangle.

From the smaller triangles we can also see that
sin θ = y / z
cos θ = (8-z) / x
So y / z = 3/5, i.e. z = 5y/3.
And (8-z) / x = 4/5, so 8-z = 4x/5.
So 8 = z + (8-z) = 5y/3 + 4x/5. Multiplying through by 15 we get
25y + 12x = 120.
So if y is the length, the area is xy = (10 - 25y/12) y = 10y - (25/12)y^2.

Conversely, using x for the length we get the area as
(6 - 12x/25)x = 6x - (12/25)x^2.

Answer 3

The easiest way to understand:

http://img171.imageshack.us/my.php?image=scan0001au4.jpg

;)