A college algebra's problem..please help :) ?
A rancher has 360 yd of fencing with which to enclose two adjacent rectangular corrals, one ofor sheep and one for cattle. a river forms one side of the carrols. suppose the width of each corral ix x yards.
a) Express the total area of the two corrals as a function of x
b) Find the domain of the function
please explain in good details, and thanx a lot !
Comments
a) Temporarily assume areas are equal, (easier to see), they are each 3 sides of fence
with a river forming the other border. Sketch this for yourself. Each area has 2 sides
of length x, so 3rd side is 180 - 2x, (if equal areas).
Single area is (180 - 2x)*x so total is twice this
A = 2x(180 - 2x) = 360x - 4x^2
If areas not equal, assign a length y to say the cattle corral.
Then A = xy + x(360 - 4x -y) = 360x - 4x^2
so this makes no difference as it is the same function for total area.
b) Domain is all about where real values can exist for y or f(x) or A in todays example.
I expect you know that this quadratic forms an upside-down parabola or a hill shape because of the minus in front of the x^2 term.
Maximum, (top of the hill), happens when
360 = 8x
x= 45 A = 8100
All values below this are "theoretically" possible for A so domain of A is from minus infinity to 8100
The domain has to be expressed in interval notation, so is given by [-infinity, 8100)
However, area is not negative so realistic, (as opposed to real), values of A are from 0 to 8100.
I hope all the explanations were what you wanted,
Regards - Ian