Multivariable Calculus 9th Edition by James Stewart solution manual

if there were no charge for each space. The ­intercept of 50 is the
smallest rental fee that results in no spaces rented.
17. (a) (b) The slope of
9
5
means that  increases
9
5
degrees for each increase
of 1 ◦ C. (Equivalently,  increases by 9 when  increases by 5
and  decreases by 9 when  decreases by 5.) The ­intercept of
32 is the Fahrenheit temperature corresponding to a Celsius
temperature of 0.
18. (a) Jari is traveling faster since the line representing her distance versus time is steeper than the corresponding line for Jade.
(b) At  = 0, Jade has traveled 10 miles. At  = 6, Jade has traveled 16 miles. Thus, Jade’s speed is
16 miles − 10 miles
6 minutes − 0 minutes
= 1 mimin. This is
1 mile
1 minute
×
60 minutes
1 hour
= 60 mih
At  = 0, Jari has traveled 0 miles. At  = 6, Jari has traveled 7 miles.Thus, Jari’s speed is
7 miles − 0 miles
6 minutes − 0 minutes
=
7
6
mimin or
7 miles
6 minutes
×
60 minutes
1 hour
= 70 mih
(c) From part (b), we have a slope of 1 (mileminute) for the linear function  modeling the distance traveled by Jade and
from the graph the ­intercept is 10. Thus, () = 1 + 10 =  + 10. Similarly, we have a slope of
7
6
milesminute for
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.
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CHAPTER 1 FUNCTIONSAND MODELS
Jari and a ­intercept of 0. Thus, the distance traveled by Jari as a function of time  (in minutes) is modeled by
() =
7
6  + 0 =
7
6 .
19. (a) Let  denote the number of chairs produced in one day and  the associated
cost. Using the points (1002200) and (3004800), we get the slope
4800−2200
300−100
=
2600
200
= 13. So  − 2200 = 13( − 100) ⇔
 = 13 + 900.
(b) The slope of the line in part (a) is 13 and it represents the cost (in dollars)
of producing each additional chair.
(c) The ­intercept is 900 and it represents the fixed daily costs of operating
the factory.
20. (a) Using  in place of  and  in place of , we find the slope to be
 2 −  1
 2 −  1
=
460 − 380
800 − 480
=
80
320
=
1
4 .
So a linear equation is  − 460 =
1
4 ( − 800)
⇔  − 460 =
1
4  − 200
⇔  =
1
4  + 260.
(b) Letting  = 1500 we get  =
1
4 (1500)+ 260 = 635.
The cost of driving 1500 miles is $635.
(c)
The slope of the line represents the cost per
mile, $025.
(d) The ­intercept represents the fixed cost, $260.
(e) A linear function gives a suitable model in this situation because you
have fixed monthly costs such as insurance and car payments, as well
as costs that increase as you drive, such as gasoline, oil, and tires, and
the cost of these for each additional mile driven is a constant.
21. (a) We are given
change in pressure
10 feet change in depth
=
434
10
= 0434. Using  for pressure and  for depth with the point
() = (015), we have the slope­intercept form of the line,  = 0434 + 15.
(b) When  = 100, then 100 = 0434 + 15 ⇔ 0434 = 85 ⇔  =
85
0434
≈ 19585 feet. Thus, the pressure is
100 lbin 2 at a depth of approximately 196 feet.
22. (a) () =  −2 and (0005) = 140, so 140 = (0005) −2 ⇔  = 140(0005) 2 = 00035.
(b) () = 00035 −2 , so for a diameter of 0008 m the resistance is (0008) = 00035(0008) −2 ≈ 547 ohms.
23. If  is the original distance from the source, then the illumination is () =  −2 =  2 . Moving halfway to the lamp gives
an illumination of   1
2 

=   1
2 
 −2
= (2) 2 = 4( 2 ), so the light is four times as bright.
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
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24. (a)  =  and  = 39 kPa when  = 0671 m 3 , so 39 = 0671 ⇔  = 39(0671) = 26169.
(b) When  = 0916,  = 26169 = 261690916 ≈ 286, so the pressure is reduced to approximately 286 kPa.