MATH 205 Quiz: MATH 205 Louisville Quiz 8 150403 Solution
MATH 205–04 Quiz #8 solutions
1. (12 points) You are enclosing an entire rectangular field with chainlink fence, then placing a
lightweight chicken-wire fence down the middle, parallel to the sides of the yard. The chainlink
fence costs $15 per foot to install, while the chickenwire will cost $2 per foot. You have $3840
for this project; what dimensions maximize the area you can enclose?
.
.x
.
x
.
y
.
y
.
y
In the above picture, we see the design drawn, with chainlink fences drawn with heavier lines.
There are five fences, and we can see that their total cost in dollars will be 15x+ 15x+ 15y+
2y+ 15y= 30x+ 32y. We are thus constrained both by the requirement that xand ybe
non-negative, and by the equation 30x+ 32y= 3840. Subject to these constraints we wish
to maximize the area, which is clearly xy. Using the given constraint, we might note that
y=3840−30x
32 = 120 −15
16 x, so the function we seek to maximize can be written as A(x) =
x(120 −15
16 x)= 120x−15
16 x2. The interval we’re maximizing on will be [0,128], because if
x > 128, then y < 0.
Note that A′(x) = 120 −15
8x, which has a critical point when 120 −15
8x= 0; simple algebra
will yield that x= 64. Thus we have three possible maximizing choices: x= 0, x= 64, and
x= 128. A(0) = A(128) = 0, while A(64) >0, so the optimum choice is to let x= 64 and
y= 120 −15
16 ·64 = 60.
2. (4 points) Using a starting point of x0= 2, show the first three steps of Newton’s method to
approximate a solution to x3−2x2+ 4 = 0; you do not need to arithmetically simplify your
result for x3.
Recall that Newton’s method is the formula xn+1 =xn−f(xn)
f′(xn). In this case, where our function
is x3−2x2+ 4, the formula will be specifically xn+1 =xn−x3
n−2x2
n+4
3x2−4x. Running through three
iterations:
x0= 2
x1= 2 −23−2·22+ 4
3·22−4·2= 2 −4
4= 1
x2= 1 −13−2·12+ 4
3·12−4·1= 1 −3
−1= 4
x3= 4 −43−2·42+ 4
3·42−4·4= 4 −36
32 =23
8
Incidentally, this Newton’s method procedure is surprisingly poor; it takes about 11 iterations
before settling down at the zero of x≈ −1.13.
Page 1 of 2 Friday, April 2, 2015
Document Summary
Quiz #8 solutions: (12 points) you are enclosing an entire rectangular eld with chainlink fence, then placing a lightweight chicken-wire fence down the middle, parallel to the sides of the yard. The chainlink fence costs per foot to install, while the chickenwire will cost per foot. You have for this project; what dimensions maximize the area you can enclose? x y y y x. In the above picture, we see the design drawn, with chainlink fences drawn with heavier lines. There are ve fences, and we can see that their total cost in dollars will be 15x + 15x + 15y + We are thus constrained both by the requirement that x and y be non-negative, and by the equation 30x + 32y = 3840. Subject to these constraints we wish to maximize the area, which is clearly xy. Using the given constraint, we might note that y = 3840 30x.
