Multivariable Calculus 9th Edition by James Stewart solution manual

25. (a)  =  3 so doubling the windspeed  gives  = (2) 3 = 8( 3 ). Thus, the power output is increased by a factor
of eight.
(b) The area swept out by the blades is given by  =  2 , where  is the blade length, so the power output is
 =  3 =  2  3 . Doubling the blade length gives  = (2) 2  3 = 4( 2  3 ). Thus, the power output is increased
by a factor of four.
(c) From part (b) we have  =  2  3 , and  = 0214 kgm 3 ,  = 30 m gives
 = 0214
kg
m 3
· 900 m 2 ·  3 = 1926 3
kg
m
For  = 10 ms, we have
 = 1926

10
m
s
 3
kg
m
= 192,600
m 2 · kg
s 3
≈ 605,000 W
Similarly,  = 15 ms gives  = 650,025 ≈ 2,042,000 W and  = 25 ms gives  = 3,009,375 ≈ 9,454,000 W.
26. (a) We graph () = (567 × 10 −8 ) 4 for 100 ≤  ≤ 300:
(b) From the graph, we see that as temperature increases, energy increases—slowly at first, but then at an increasing rate.
27. (a) The data appear to be periodic and a sine or cosine function would make the best model. A model of the form
() = cos() +  seems appropriate.
(b) The data appear to be decreasing in a linear fashion. A model of the form () =  +  seems appropriate.
28. (a) The data appear to be increasing exponentially. A model of the form () =  ·   or () =  ·   +  seems
appropriate.
(b) The data appear to be decreasing similarly to the values of the reciprocal function. A model of the form () = 
seems appropriate.
c ° 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.
28
¤
CHAPTER 1 FUNCTIONSAND MODELS
Exercises 29 – 33: Some values are given to many decimal places. The results may depend on the technology used — rounding is left to the reader.
29. (a)
A linear model does seem appropriate.
(b) Using the points (4000141) and (60,00082), we obtain
 − 141 =
82 − 141
60,000 − 4000
( − 4000) or, equivalently,
 ≈ −0000105357 + 14521429.
(c) Using a computing device, we obtain the regression line  = −00000997855 + 13950764.
The following commands and screens illustrate how to find the regression line on a TI­84 Plus calculator.
Enter the data into list one (L1) and list two (L2). Press to enter the editor.
Find the regession line and store it in Y 1 . Press .
Note from the last figure that the regression line has been stored in Y 1 and that Plot1 has been turned on (Plot1 is
highlighted). You can turn on Plot1 from the Y= menu by placing the cursor on Plot1 and pressing or by
pressing .
Now press to produce a graph of the data and the regression
line. Note that choice 9 of the ZOOM menu automatically selects a window
that displays all of the data.
(d) When  = 25,000,  ≈ 11456; or about 115 per 100 population.
c ° 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.
SECTION 1.2 MATHEMATICAL MODELS:A CATALOG OF ESSENTIALFUNCTIONS
¤
29
(e) When  = 80,000,  ≈ 5968; or about a 6% chance.
(f) When  = 200,000,  is negative, so the model does not apply.
30. (a) Using a computing device, we obtain the regression line  = 001879 + 030480.
(b) The regression line appears to be a suitable model for the data.
(c) The ­intercept represents the percentage of laboratory rats that
develop lung tumors when not exposed to asbestos fibers.
31. (a) (b) Using a computing device, we obtain the regression line
 = 188074 + 8264974.
(c) When  = 53 cm,  ≈ 1823 cm.
32. (a) See the scatter plot in part (b). A linear model seems appropriate.
(b) Using a computing device, we obtain the regression line
 = 031567 + 815578.
(c) For 2005,  = 5 and  ≈ 973 centskWh. For 2017,  = 17 and
 ≈ 1352 centskWh.
33. (a) See the scatter plot in part (b). A linear model seems appropriate.
(b) Using a computing device, we obtain the regression line
 = 112486 + 60,11986.
(c) For 2002,  = 17 and  ≈ 79,242 thousands of barrels per day.
For 2017,  = 32 and  ≈ 96,115 thousands of barrels per day.
34. (a)  = 1000431227 1499 528750