11 11 11 33 11
112 sin(42 sin(4 x x )) 2 sin 4
2 sin 4
x x 44
sin 4
sin 4
x x 44
3 3
2 2
2
4 2
4
x x 44
2 2 3
3 8 8
4 4
8
x 8
x 44
4 4
8
8
88
44
Step 3
Step 3 Make a table of values.Make a table of values.
Steps 4 and 5
Steps 4 and 5 Plot the points found in the table and join them with a sinu-Plot the points found in the table and join them with a sinu-soidal curve. Figure
soidal curve. Figure 31 shows the graph,31 shows the graph, extended to the rigextended to the right and leftht and left to include two full periods.
to include two full periods.
Now try Exercise 49.
Now try Exercise 49.
Determining a Trigonometric Model Using Curve Fitting
Determining a Trigonometric Model Using Curve Fitting
A sinu-Asinu-soidal function is often a good approximation of a set of real data points.
soidal function is often a good approximation of a set of real data points.
EXAMPLE 8
EXAMPLE 8 Modeling Temperature with a Sine FunctionModeling Temperature with a Sine Function
The maximum average monthly temperature in New Orleans is 82
The maximum average monthly temperature in New Orleans is 82 °°F and theF and the minimum is 54
minimum is 54°°FF. The . The table shows the average monthly table shows the average monthly temperatures. The scat-temperatures. The scat-ter diagram for a 2-year inscat-terval in Figure 32 strongly suggests that the ter diagram for a 2-year interval in Figure 32 strongly suggests that the tempera-tures can be modeled with a sine curve.
tures can be modeled with a sine curve.
(a)
(a) Using only the maximum Using only the maximum and minimum temperand minimum temperatures,atures, determine a functiondetermine a function of
of the the form form wherewhere aa,, bb,, cc,, aanndd d d are constants,are constants, that models the average monthly temperature in New Orleans. Let
that models the average monthly temperature in New Orleans. Let x x repre- repre-sent the
sent the month,month, with January with January corresponding tocorresponding to x x 1.1.
f
f
x x
aa sinsin
bb
x x d d
cc,,c = c = – –11 y
y
x 0 x
0 1 1
– –22 – –33
y
y==––1 + 2 sin(41 + 2 sin(4 x x++ )) –
–11 – –
4 – 4 –
2 2
4 4
2 2
8 8
3 3
8 – 8
–33
8 8
8 –8 –
Figure 31 Figure 31
85 85
50 50 1
1 2525
Figure 32 Figure 32 Month
Month FF MMoonntthh FF JJaann 5544 JJuullyy 8822 F
Feebb 5555 AAuugg 8811 M
Maarr 6611 SSeepptt 7777 A
Apprr 6699 OOcctt 7711 M
Maayy 7733 NNoovv 5599 JJuunnee 7799 DDeecc 5555
Source:
Source: Miller, A., J. Thompson, and R.Miller, A., J. Thompson, and R.
Peterson,
Peterson, Elements of Meteorology, 4th Elements of Meteorology, 4th Edition,
Edition, Charles E. Merrill PublishingCharles E. Merrill Publishing Co., 1983.
Co., 1983.
6.3
6.3 Graphs of the Sine and Graphs of the Sine and Cosine FunctionsCosine Functions 567567
(b)
(b) On the same On the same coordinate axcoordinate axes,es, graphgraph f f for a two-year period together with thefor a two-year period together with the actual data values found in the
actual data values found in the table.table.
(c)
(c) Use theUse the sine regressionsine regression feature of a graphing calculator to determine a sec-feature of a graphing calculator to determine a sec-ond model for these data.
ond model for these data.
Solution Solution (a)
(a) We use the maximum and minimum average monthly temperatures toWe use the maximum and minimum average monthly temperatures to fifindnd the amplitude
the amplitude aa..
The average of the maximum and minimum temperatures is a good choice The average of the maximum and minimum temperatures is a good choice for
for cc. The average is. The average is
Since
Since the the coldest coldest month month is is JanuaryJanuary,, when when and and the the hottest hottest month month isis July
July,, when when we we shoulshould d chooschoosee d d to be about 4. We experiment withto be about 4. We experiment with values just greater than 4 to
values just greater than 4 to fifindnd d d . Trial and error using a . Trial and error using a calculator leads tocalculator leads to Since temperatures repeat every 12
Since temperatures repeat every 12 months,months, bb is is Thus,Thus,
(b)
(b) Figure Figure 33 33 shows shows the the data data points points and and the the graph graph of of forfor comparison. The horizontal translation of the model is fairly obvious here.
comparison. The horizontal translation of the model is fairly obvious here.
y
y 14 sin14 sin 66 x x 6868 f
f
x x
aa sinsin
bb
x xd d
cc 14 sin14 sin
66
x x 4.24.2
22 68.68.
12 12
6 6..
d d 4.2.4.2.
x x 7,7,
x x1,1, 82
825454 2 2
68.68.
a
a 8282 5454 2
2 1414
((aa)) ((bb))
FFiigguurree3333 FFiigguurree3344 90 90
50 50 1
1 2525
V
Values are rounded to alues are rounded to thethe nearest hundredth.
nearest hundredth.
90 90
50 50 1
1 2525
(c)
(c) We used the given data for a two-year period to produce the model de-We used the given data for a two-year period to produce the model de-scribed in Figure 34(a). Figure 34(b) shows its graph along with the data scribed in Figure 34(a). Figure 34(b) shows its graph along with the data points.
points.
Now try Exercise 73.
Now try Exercise 73.
568
568 CHAPTER 6CHAPTER 6 The Circular Functions and Their Graphs The Circular Functions and Their Graphs
Concept Check
Concept Check In Exercises 1 In Exercises 1 – – 4, match each function with its graph.4, match each function with its graph.
1
Concept Check In Exercises 21 and 22, give the equation of a sine function having the In Exercises 21 and 22, give the equation of a sine function having the given graph.
6.3 Exercises Exercises
6.3
6.3 Graphs of the Sine and Graphs of the Sine and Cosine FunctionsCosine Functions 569569
Concept Check
Concept Check Match each function in Column I with the appropriate description in Match each function in Column I with the appropriate description in Column II.
Concept Check Match each function with its graph. Match each function with its graph.
27.
Find the amplitude, the period, any vertical translation, and any phase shift of the graph Find the amplitude, the period, any vertical translation, and any phase shift of the graph of each function. See Examples 5
of each function. See Examples 5 – – 7.7.
3
Graph each function over a two-period interval. See Example 5.
Graph each function over a two-period interval. See Example 5.
3
377.. 3388..
3
399.. 4400..
Graph each function over a one-period interval.
Graph each function over a one-period interval.
4
411.. 4422..
4
433.. 4444..
Graph each function over a two-period interval. See Example 6.
Graph each function over a two-period interval. See Example 6.
4
There are other correct answers in There are other correct answers in Exercises 21 and 22.
Exercises 21 and 22.
21.
570
570 CHAPTER 6CHAPTER 6 The Circular Functions and Their Graphs The Circular Functions and Their Graphs
4
477.. 4488..
Graph each function over a one-period interval. See Example 7.
Graph each function over a one-period interval. See Example 7.
4
499.. 5500..
5
511.. 5522..
Concept Check
Concept Check In Exercises 53 and 54, In Exercises 53 and 54, fi find the equation of a sine function having thend the equation of a sine function having the given graph.
(Modeling) Solve each problem.Solve each problem.
55.
55. Aver Average age AnnualAnnualTemperTemperatureature Scientists believe that the average Scientists believe that the average annual tempera-annual tempera-ture in a given location is periodic. The average temperatempera-ture at a
ture in a given location is periodic. The average temperature at a given place duringgiven place during a given season
a given season flfluctuates as time goes on, from colder to warmer, uctuates as time goes on, from colder to warmer, and back to cold-and back to cold-er. The graph shows an idealized description of the temperature (in
er. The graph shows an idealized description of the temperature (in F) for the lastF) for the last few thousand years of a location at the same
few thousand years of a location at the same latitude as Anchorage, Alaska.latitude as Anchorage, Alaska.
(a)
(a) Find the highest and Find the highest and lowest temperatures recorded.lowest temperatures recorded.
(b)
(b) Use these two numbers toUse these two numbers to fifind the amplitude.nd the amplitude.
(c)
(c) Find the period of the Find the period of the function.function.
(d)
(d) What is the trend of What is the trend of the temperature now?the temperature now?
Years ago Years ago
Average Annual Temperature (Idealized) Average Annual Temperature (Idealized)
1
There are other correct answers in There are other correct answers in Exercises 53 and 54.
Exercises 53 and 54.
53.
(c)about 35,000 yrabout 35,000 yr (d)(d)downwarddownward y
56.
56. Blood Pressure Variation Blood Pressure Variation The graph gives the variation in blood pressure for aThe graph gives the variation in blood pressure for a typical person. Systolic and diastolic pressures are the upper and lower limits typical person. Systolic and diastolic pressures are the upper and lower limits of theof the periodic changes in pressure that produce the pulse. The length of time between periodic changes in pressure that produce the pulse. The length of time between peaks is called the period of the pulse.
peaks is called the period of the pulse.
(a)
(a) Find the amplitude of the graph.Find the amplitude of the graph.
(b)
(b) Find the pulse rate (the number of pulse beats in 1 min) for this person.Find the pulse rate (the number of pulse beats in 1 min) for this person.
57.
57. Activity of a Nocturnal Animal Activity of a Nocturnal Animal Many of the activities of living organisms areMany of the activities of living organisms are periodic. For example, the graph below shows the time that a
periodic. For example, the graph below shows the time that a certain nocturnal ani-certain nocturnal ani-mal begins its evening activity.
mal begins its evening activity.
(a)
(a) Find the amplitude of this graph.Find the amplitude of this graph. (b)(b) Find the period.Find the period.
58.
58. Position of a Moving Arm Position of a Moving Arm TheThe fifigure shows schematic diagrams of a rhythmicallygure shows schematic diagrams of a rhythmically moving arm. The upper arm
moving arm. The upper arm RO ROrotates back and forth about the pointrotates back and forth about the point R R; the position; the position of the arm is measured by the angle
of the arm is measured by the angle y ybetween the actual position and the downwardbetween the actual position and the downward vertical position. (
vertical position. (Source:Source: De Sapio, Rodolfo,De Sapio, Rodolfo, Calculus for the Life Sciences.Calculus for the Life Sciences.
Copyright ©
Copyright © 1978 by W. H. Freeman and Company. Rep1978 by W. H. Freeman and Company. Reprinted by permission.)rinted by permission.)
(a)
(a) Find Find an an equation equation of of the the form form for for the the graph graph shown.shown.
(b)
(b) How long does it take for a complete movement of the arm?How long does it take for a complete movement of the arm?
y
yaasinsinkt kt
3 3_ _ 2 2 y
y
t t
A A n n g g l l e e o o f f a a r r m m y y , ,
Time, in seconds, Time, in seconds, t t This graph shows the relationship This graph shows the relationship between angle
between angle y y and timeand time t t in seconds.in seconds.
0 0 1 1_ _3 –3 – 1 1_ _ 3 3
y y O O
R R
y y
R R
O O
R
R RR RR RR
OO OO OO OO R
R
O O y y
A
Apprr JJuunn AAuugg OOcctt DDeecc FFeebb AApprr 6:30
6:30 7:00 7:00 7:30 7:30 8:00 8:00
4:00 4:00 4:30 4:30 5:00 5:00 5:30 T5:30 T i i P P M M m m e e . . . .
Month Month
Activity of a Nocturnal Animal Activity of a Nocturnal Animal
Time (in seconds) Time (in seconds) Blood Pressure Variation Blood Pressure Variation
P P r r e e s s s s u u r r e e ( ( i i n n m m m m m m e e r r c c u u r r y y ) )
..88 11..66 Systolic
Systolic pressure pressure
Diastolic Diastolic pressure pressure 80 80
40 40
0 0 120 120
Period =.8 sec Period =.8 sec
6.3
6.3 Graphs of the Sine and Graphs of the Sine and Cosine FunctionsCosine Functions 571571
5
566.. ((aa))2020 (b)(b)7575 5
577.. ((aa))about 2 hrabout 2 hr (b)(b)1 yr1 yr 5
588.. ((aa)) ((bb)) 33 secsec 2 y 2
y 11 3
3 sinsin44 t t 3 3
572
572 CHAPTER 6CHAPTER 6 The Circular Functions and Their Graphs The Circular Functions and Their Graphs
Tides for Kahului Harbor
Tides for Kahului Harbor The chart shows the tides for Kahului Harbor (on the island The chart shows the tides for Kahului Harbor (on the island of Maui, Hawaii). To identify high and low tides and times for other Maui areas, the of Maui, Hawaii). To identify high and low tides and times for other Maui areas, the fol-lowing adjustments must be made.
lowing adjustments must be made.
59.
59. 24 hr24 hr 60.
60. approximatelyapproximately 61.
61. approximately 6:00approximately 6:00PP..MM.;.;
approximately .2 ft approximately .2 ft 62.
62. approximately 7:19approximately 7:19PP..MM.;.;
approximately 0 ft approximately 0 ft 63.
63. approximately 2:00approximately 2:00AA..MM.;.;
approximately 2.6 ft approximately 2.6 ft 64.
64. approximately 3:18approximately 3:18AA..MM.;.;
approximately 2.4 ft approximately 2.4 ft 65.
65. 1; 2401; 240oror
66.
66. 1; 1201; 120oror
6
677.. ((aa))5;5; (b)(b)6060 (c)
(c)5; 1.545;5; 1.545;4.045;4.045;4.045;4.045;
1.545 1.545 (d)(d)
0
0 t t
E E
– –55
5 5
.
.00116677 ..003333 .025 .025 .0083 .0083
E
E= 5 cos 120= 5 cos 120 t t
1 1 60 60
2 2
3 3 4 4
3 3
2.6 2.6.2.2
2
2 1.21.2
H
Haannaa:: HHiigghh,, 40 min,40 min,.1 ft;.1 ft;
Low,
Low,18 min,18 min,.2 ft.2 ft
Ma
Makekenana:: HiHighgh,,1:21,1:21,.5 ft;.5 ft;
Low,
Low, 1:09,1:09,.2 ft.2 ft Maa
Maalaelaea:a: HiHigh,gh,1:52,1:52,.1 ft;.1 ft;
Low,
Low,1:19,1:19,.2 ft.2 ft
La
Lahahainina:a: HiHighgh,,1:18,1:18,.2 ft;.2 ft;
Low,
Low, 1:01,1:01,.1 ft.1 ft
Use the graph to work Exercises 59 Use the graph to work Exercises 59 – – 64.64.
59.
59. The graph is an example of The graph is an example of a periodic function. What is the period (in hours)?a periodic function. What is the period (in hours)?
60.
60. What is the What is the amplitude?amplitude?
61.
61. At what time on January 20 was low tide at KahuluAt what time on January 20 was low tide at Kahului?i? What was the height?What was the height?
62.
62. Repeat Exercise 61 for Maalaea.Repeat Exercise 61 for Maalaea.
63.
63. At what time on January 22 was high tide at KahulAt what time on January 22 was high tide at Kahului?ui? What was the height?What was the height?
64.
64. Repeat Exercise 63 for Lahaina.Repeat Exercise 63 for Lahaina.
Musical Sound Waves
Musical Sound Waves Pure sounds produce single sine waves on an oscilloscope.Pure sounds produce single sine waves on an oscilloscope.
Find the amplitude and period of each sine wave graph in Exercises 65 and 66. On the Find the amplitude and period of each sine wave graph in Exercises 65 and 66. On the vertical scale, each square represents
vertical scale, each square represents.5.5; on the horizontal scale, each square represents; on the horizontal scale, each square represents 30
30 or or ..
6
655.. 6666..
(Modeling)
(Modeling) Solve each problem.Solve each problem.
67.
67. Voltage of an Electrical CircuitVoltage of an Electrical Circuit The voltageThe voltage E E in an electrical circuit is modeled byin an electrical circuit is modeled by ,,
where
wheret t is time measured in seconds.is time measured in seconds.
(a)
(a) Find the amplitude and the period.Find the amplitude and the period.
(b)
(b) How many cycles are completed in 1 sec? (The number of cycles (periods)How many cycles are completed in 1 sec? (The number of cycles (periods) completed in 1 sec is the
completed in 1 sec is the frequency frequencyof the function.)of the function.) (c)
(c) FindFind E E when when , , .03, .03, .06, .06, .09, .09, .12..12.
(d)
(d) GraphGraph E E for for 00t t 303011.. t
t 00
E
E 5 cos 1205 cos 120 t t
6 6
JANUARY JANUARY
1
19 9 2 20 0 2 21 1 2 22 2
6 6 am amNoNoonon 66
pm pm 6
6 am amNoNoonon 66
pm pm 6
6 am amNoNoonon 66
pm pm 6
6 am amNoNoonon 66
pm pm
0 0 1 1 2 2 3 3
0 0 1 1 2 2 3 3
F F e e e e t t
F F e e e e t t
Source:
Source: Maui News Maui News. Original chart prepared by. Original chart prepared by Edward K. Noda and Associates.
Edward K. Noda and Associates.
6.3
6.3 Graphs of the Sine and Graphs of the Sine and Cosine FunctionsCosine Functions 573573
68.
68. Voltage of an Electrical CircuitVoltage of an Electrical Circuit For another electrical circuit, the voltageFor another electrical circuit, the voltage E E isis modeled by
modeled by
,, where
wheret t is time measured in seconds.is time measured in seconds.
(a)
(a) Find the amplitude and the period.Find the amplitude and the period.
(b)
(b) Find the frequency. See Exercise 67(b).Find the frequency. See Exercise 67(b).
(c)
(c) FindFind E E when when , .04, , .04, .08, .08, .12, .12, .14..14.
(d)
(d) Graph one period of Graph one period of E E ..
69.
69. Atmospheric Atmospheric Carbon Carbon DioxideDioxideAtAt Mauna Loa, Hawaii, atmospheric carbon Mauna Loa, Hawaii, atmospheric carbon dioxide levels in parts per million (ppm) dioxide levels in parts per million (ppm) have been measured regularly since 1958.
have been measured regularly since 1958.
The function de
The function defifined byned by
can be used to model these levels, where can be used to model these levels, where x
x is is in in years years and and corresponds corresponds toto 1960. (
1960. (Source:Source: Nilsson, A.,Nilsson, A., GreenhouseGreenhouse Earth,
Earth,John WiJohn Wiley &ley & Sons, 19Sons, 1992.)92.) (a)
(a) GraphGraph L Lin in the the window window by by ..
(b)
(b) When do the seasonal maximum and minimum carbon dioxide levels occur?When do the seasonal maximum and minimum carbon dioxide levels occur?
(c)
(c) L Lis the sum of a is the sum of a quadratic function and a sine function. What is the signiquadratic function and a sine function. What is the signi fificancecance of each of these functions?
of each of these functions? Discuss what physical phenomena may Discuss what physical phenomena may be responsiblebe responsible for each function.
for each function.
70.
70. Atmos Atmospheripheric c CarbonCarbonDioxidDioxidee Refer to Exercise 69. The Refer to Exercise 69. The carbon dioxide content incarbon dioxide content in the atmosphere at Barrow, Alaska, in parts per million (ppm) can be modeled using the atmosphere at Barrow, Alaska, in parts per million (ppm) can be modeled using the function de
the function defifined byned by
,, where
where corresponds corresponds to to 1970. 1970. ((Source:Source:Zeilik, M. and S. Gregory,Zeilik, M. and S. Gregory, Introductory Introductory Astronomy and Astrophysics,
Astronomy and Astrophysics,Brooks/Cole, 1998.)Brooks/Cole, 1998.) (a)
(a) GraphGraphC C in in the the window window by by ..
(b)
(b) Discuss possible reasons why the amplitude of the Discuss possible reasons why the amplitude of the oscillations in the graph of oscillations in the graph of C C is larger than the amplitude of the oscillations in the graph of
is larger than the amplitude of the oscillations in the graph of L Lin Exercise 69,in Exercise 69, which models Hawaii.
which models Hawaii.
(c)
(c) DeDefifine a new functionne a new function C C that is valid if that is valid if x x represents the actual year, whererepresents the actual year, where ..
71.
71. Temperature in FairbanksTemperature in Fairbanks The temperature in Fairbanks is modeled byThe temperature in Fairbanks is modeled by ,,
where
where is is the the temperature temperature in in degrees degrees Fahrenheit Fahrenheit on on dayday x x, , with with corre- corre-sponding
sponding to to January January 1 1 and and corresponding corresponding to to December December 31. 31. Use Use a a calculatorcalculator to estimate the temperature on the following days. (
to estimate the temperature on the following days. (Source:Source:Lando, B. and C. Lando,Lando, B. and C. Lando,
““Is the Graph of Is the Graph of Temperature Variation a Sine Curve?Temperature Variation a Sine Curve?””,,The Mathematics Teacher,The Mathematics Teacher, 70, September 1977.)
70, September 1977.) (a)
(a) March 1 (day 60)March 1 (day 60) (b)(b) April 1 (day 91)April 1 (day 91) (c)(c) Day 150Day 150 (d)
(d) June 15June 15 (e)(e) September 1September 1 (f)(f) October 31October 31 72.
72. Fluctuation in the Solar Constant Fluctuation in the Solar Constant TheThesolar constant Ssolar constant S is the amount of energyis the amount of energy per unit area that reaches Earth
per unit area that reaches Earth’’s atmosphere from the sun. It is equal to 1367 wattss atmosphere from the sun. It is equal to 1367 watts x
x365365
x x11 T
T
x x
T
T
x x
37 sin37 sin
36536522
x x101101
25251970
1970 x x19951995
320,380320,380
5,255,25
x x00
C
C
x x
.04.04 x x22.6.6 x x3303307.5 sin7.5 sin
22 x x
325,365325,365
15,3515,35
x x00 L
L
x x
.022.022 x x22.55.55 x x3163163.5 sin3.5 sin
22 x x
t t .02.02
E
E 3.8 cos 403.8 cos 40 t t 6
688.. ((aa))3.8;3.8; (b)(b)2020
(c)
(c)3.074; 1.174;3.074; 1.174;3.074;3.074;
3.074; 1.1743.074; 1.174 (d)
(d)
6 699.. ((aa))
(b)
(b)maximums: maximums: , , ,,
;
; minimums: minimums: , , ,,
7 700.. ((aa))
(c) (c)
7
711.. ((aa))11 (b)(b)1919 (c)(c)5353 (d)(d)5858
(e)
(e)4848 (f(f))1212
7.5 sin
7.5 sin
22
x x19701970
.6
.6
x x19701970
330330C
C
x x
.04.04
x x19701970
22 380380
320 320 5
5 2525
C
C (( x x) = .04) = .04 x x22+ .6+ .6 x x+ 330+ 330 + 7.5 sin(2 + 7.5 sin(2 x x)) 11
11 4 4 , . . ., . . .
7 7 4 x 4
x 33 4 4 9
9 4 4 , . . ., . . .
5 5 4 x 4
x 11 4 4 365
365
325 325 1
155 3355
L
L(( x x) = .022) = .022 x x22+ .55+ .55 x x+ 316+ 316 + 3.5 sin(2
+ 3.5 sin(2 x x))
0
0 t t
E E
– –3.83.8
3.8 3.8
. .002255 ..0055
E
E= 3.8 cos 40= 3.8 cos 40 t t
1 1 20 20
9 9
9
9
95
574 CHAPTER 6CHAPTER 6 The Circular Functions and Their Graphs The Circular Functions and Their Graphs
per square meter
per square meter but varies slightly throughout the seasons. Thisbut varies slightly throughout the seasons. This flfluctuation uctuation ininSS can be calculated using the formula
can be calculated using the formula
..
In this formula,
In this formula, N N is is the the day day number number covering covering a a four-year four-year period, period, where where cor- cor-responds
responds to to January January 1 1 of of a a leap leap year year and and corresponds to corresponds to December December 31 31 of of the fourth year. (
the fourth year. (Source:Source: Winter, C., R. Sizmann, and Vant-Hunt (Editors),Winter, C., R. Sizmann, and Vant-Hunt (Editors), Solar Solar Power Plants,
Power Plants,Springer-Verlag, 1991.)Springer-Verlag, 1991.) (a)
(Modeling) Solve each problem. See Example 8.Solve each problem. See Example 8.
73.
73. Average Monthly Temperature Average Monthly Temperature The averageThe average monthly temperature (in
monthly temperature (in F) in Vancouver,F) in Vancouver, Canada, is shown in the table.
Canada, is shown in the table.
(a)
(a) Plot the average monthly temperaturePlot the average monthly temperature over a two-year period letting
over a two-year period letting
correspond to the month of January correspond to the month of January during the
during the fifirst year. Do the data seemrst year. Do the data seem to indicate a translated sine graph?
to indicate a translated sine graph?
(b)
(b) The highest average monthly temperatureThe highest average monthly temperature is 64
is 64F in July, and the lowest averageF in July, and the lowest average
is 64F in July, and the lowest averageF in July, and the lowest average