Page 140 - Calculus Workbook For Dummies

124 Part III: Differentiation

Q. A rancher has 400 feet of fencing and wants 3. Determine the domain of the function.
to build a corral that’s divided into three You can’t have a negative length of fence,
equal rectangles. See the following figure. so x can’t be negative. And if you build
What length and width will maximize the the ridiculous corral with no width, all
area? 400 feet of fencing would equal 6x. So
x
x x x x $ 0 and 6 # 400
200
x #
3
y y y y
4. Find the critical numbers of A xh.
^
5
A x = 300 x - 4 . x 2
^ h
l ^ h
x x x A x = 300 - 9 x
0 = 300 - 9 x
A. 100 feet by 50 feet with an area of 5000 9 x = 300
square feet.
x = 100
1. Draw a diagram and label with 3
variables. 100
A xh is defined everywhere, so 3 is
l ^
2. a. Express the thing you want maxi- the only critical number.
mized, the area, as a function of the 5. Evaluate A xh at the critical number
variables. ^
and at the endpoints of the domain.
Area Length Width
#
=
A 0 = 0
^ h
A 3 $
=
x y
100 100 100 2
b. Use the given information to relate A c 3 m = 300c 3 m - . 4 5 c 3 m
the two variables to each other. = 5000
6 x + y 4 = 400 200
A c m = 0
3 x + y 2 = 200 3
c. Solve for one variable and substitute The first and third results above should
into the equation from Step 2a to be obvious because they represent cor-
create a function of a single variable. rals with zero length and zero width.
100
You’re done. x = will maximize the
y 2 = 200 - 3 x 3
5
area. Plug that into y = 100 - 1 . x and
5
y = 100 - 1 . x
you get y = 50. So the largest corral is
A 3 $ 100
x y
=
3 $ , or 100 feet long, 50 feet wide,
5
x 100 -
A x = 3 ^ 1 . xh 3
^ h
and has an area of 5000 square feet.
= 300 x - 4 . x 2
5