Transcript Slide 1

Homework
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Homework Assignment #27
Review Section 4.6
Page 265, Exercises: 1 – 29 (EOO),19, skip
17
Quiz next time
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 265
1.
Find the dimensions of the rectangle of maximum area
that can be formed from a 50-in. piece of wire.
(a)
What is the constraint equation relating the lengths x and
y of the two sides?
2x + 2y = 50
(b)
Find a formula for the area in terms of x alone.
A  xy  2 x  2 y  50  x  y  25  y  25  x  A  x  25  x 
(c)
Does the problem require optimization over an open
interval or a closed interval?
The problem requires optimization on a closed area.
(d)
Solve the optimization problem.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 265
1.
(d)
Solve the optimization problem.
A  x  25  x   A  25  2 x  A  0  25  2 x  x  12.5
A  12.5  25  12.5  156.25  Dimensions are 12.5 by 12.5.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 265
5.
Find the positive numbers x and y such that xy = 16 and
x + y is as small as possible.
xy  16  y 
16
16
16
 s  x   s  1  2
x
x
x
16
16
x2
s  0  1  2  2  1 
 1  x 2  16  x  4
16
x
x
16
x y  4  44 8
4
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 265
9.
Suppose 600 ft. of fencing are used to enclose a corral in the
shape of a rectangle with a semicircle whose diameter is a side of the
rectangle. Find the dimensions of the corral with maximum area.
1  w
 w2
A  lw      lw 
2 2
8
w
P  2l  w    600
2
w
600  w  
w w
2
l
 300  
2
2
4
w w
 w2
w2  w2

A   300  
 300 w 

w
2
4 
8
2
8

2
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 265
9.
Continued.
w

300

A  300  w 
 0  w 1    300  w 
 168.030
4
 
 4
1  
 4
168.030  168.030
l  300 

 84.015  168.030 ft  84.015 ft
2
4
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 265
13.
Find the point P on the parabola y = x2 closest to the point
(3, 0).
y  x2
d
3  x    y  0
2
2
 9  6x  x   x
d  x4  x2  6x  9  d  
d 
2 x3  x  3
2

2 2
4 x3  2 x  6
2 x4  x2  6x  9
 d   0, 2 x 3  x  3  0
x4  x2  6x  9
d   0, x  1  Closest point to  3, 0  is 1,1 .
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 265
19. The volume of a right circular cone is

3
r 2 h and its surface area is
S   r r 2  h 2 . Find the dimensions of the cone with surface area 1
and maximal volume.
V

3
r 2 h, S   r r 2  h 2  1  1   2 r 2  r 2  h 2 
2 4
1


r
1   2r 4   2r 2h2   2r 2h2  1   2r 4  h2 
 2r 2
1   2r 4
 2 1   2r 4 r
2 4
h

V

r

1


r
2 2
2 2
 r
3
 r
3
2 3

 1  2 2 r 4  1   2 r 4 
r

4

r 
1 
V  
 1   2 r 4 1   

2 4
 3
3  2 1   2r 4
1  r



Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 265
19.
Continued.
2 3

 1  2 2 r 4  1   2 r 4 
r

4

r 


V  
 1   2 r 4 1   

2 4
 3
3  2 1   2r 4
1  r



V   0  1  3 2 r 4  0  r 4 
1 
h
2
1
r
3
1
4

2
1
2
3
3 2 
1
1

2
3
3
1
3
2
r
4
1
3
2
1

3
1
4

1
2
2
3 
 1
1
 3 34 
3
2
,h 
3
1
4

Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 265
21. What are the dimensions of the cylinder of the largest volume
that can be inscribed in the sphere of radius R?
V   r 2h 

h
 R2  r 2
2

V   r 2 2 R 2  r 2  2 r 2 R 2  r 2

2r
V   2 r 
2
2
2
R

r

2

2
2

2

R

r
 2r 


  r 3  2r  R 2  r 2  

 2 
2
2


R

r


Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 265
21. Continued.
V   0   r 2  2  R 2  r 2   0  2 R 2  3r 2  0
2R2
2
h
r

R 
3
3
2
R2 2R2
2R

h
1
3
3
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 265
25.
Consider a rectangular warehouse consisting of n separate
spaces of equal size. Assume that wall materials cost $200 per linear
ft and the company allocates $2,400,000 for the project. Find a
formula in terms of n for the maximum possible area of the warehouse.
A  lw, P  2l   n  1 w  200 P  2, 400, 000  P  12, 000
 12, 000   n  1 w 
2l  12, 000   n  1 w  A  
w
2


12, 000w   n  1 w2
A
 A  6, 000   n  1 w  A  0
2
6, 000
6, 000
6, 000   n  1 w  w 
 2l  12, 000   n  1
n 1
n 1
18, 000, 000
l  3, 000  A  lw 
n 1
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 265
29.
Find the maximum area of a triangle in the first quadrant formed
by the x-axis, the y-axis, and the tangent to y = (x + 1)–2
y   x  1  y  2  x  1
2
3
Rectangle with largest area for given perimeter is a square.
Triangle with largest area for given perimeter is 45  45  90
1
1
3
3
3
y  1  2  x  1  

x

1

2

x

2  1  0.260
3
2  x  1
y

3
 
2 1 1
2
 0.630  y  0.630    x  0.260 
1
1
y   x  0.890  A  xy  0.890 0.890  0.396
2
2
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 265
32.
An 8-billion bushel corn crop brings a price of $2.40/bushel.
A commodity broker uses the rule of thumb: If the crop of reduced by
x percent, then the price increases by 10x cents. Which crop size
results in the maximum revenue and what is the price per bushel?
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 265
36.
Figure 21 shows a rectangular plot of size 100 X 200 ft. Pipe
is to be laid between points A and C by way of point P. The cost of
laying pipe through the lot is $30/ft ant the cost along the side is $15/ft.
(a) Let f (x) be the total cost, where x is the distance from P to B.
Determine f (x), noting f is discontinuous at x = 0.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 265
36.
(b) What is the most economical way to lay the pipe? What if
the cost along the side is $24/ft?
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework
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Homework Assignment #28
Read Section 4.7
Page 265, Exercises: 31, 39, 43, 47, 53, 59
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company