Math Help: March 2008

Thursday, March 27, 2008

Wednesday, March 26, 2008

Make the equation (I am not sure if I rember all the detales exactly for this one , I am also not sure if it starts at zero , so if u can show mw what wouldt be the differance if it starts at zero or not)
A water flow reaches a min of 20 m and a max of 30 m , it takes it 6.2 hours to complete a full cycle it starts when t=0. write a sine and a cosine function to model the wave.
b) how long does it take to reach 28 m?

Range 30-20 is 10 so the amplitude is 5

Period is 6.2

Frequency is 2pi/6.2

Beginning point (0, 25)

y=____sin(____x+____)+____

y=5 sin(2pi/6.2 x ) +25

y=5 cos(2pi/6.2 x - pi/2) +25

To make it cos you need a phase shift of right of pi/2 (this is always true to go between sin and cos it doesn't matter what the numbers are)

To reach 28 m

Plug in 28 for y

28 = y=5 sin(2pi/6.2 x ) +25

subtract 25

3 = 5 sin(2pi/6.2 x )

divide by 5

0.6 = sin(2pi/6.2 x )

sin inverse of (0.6) = 2pi/6.2 x

.64 = 2pi/6.2 x

x = .63

So a little over half an hour.

YOU CAN SEE THIS ON THE GRAPH ABOVE

The final problem that I remember is
A wheel has a diameter of 70 and it is spins at 30 km/h
what is the angular velocity in m/s?

Sarah Graphing

-3 cos (x-pi/4)

Amplitude -3 so it is flipped upside down and 3 tall

then phase shift pi/4 to the right

4 sin(pi/2*x-pi/2)-1

Notice this has a phase shift of -pi/2 so it moves right

Frequency is pi/2 so the period is 2 pi divided by pi/2 (6.28/1.57)

Frequency is 4.

If you look at the graph you can see that one cycle completes at 4.

Prove the identites:
2tanx/1+tan squaredx=sin2x

2tanx/sec^2 x

2 tanx * cos^2 x

2 (sinx/cosx) *cos^2 x

2 sinx * cosx

sin2x

Cotx+tanx=2csc2x

cosx/sinx + sinx/cosx

cosxcosx/sinxcosx + sinxsinx/sinxcosx

(cos^2 x +sin^2 x)/(sinxcosx)

1/sinxcosx (since 2sinxcosx= sin2x then sinxcosx = (1/2) sin(2x) )

2/sin(2x)

2csc(2x)

Solve for x
sin squaredx - cos squaredx=1/2

-cos^2 x + sin^2 x = 1/2 Use double angle formula

2cos^2 x-1= -1/2

2cos^2 x= 1/2

cos^2 x = 1/4

cos x = +or- 1/2

x = 2.09 or 1.05

2 sin squaredx - 3 cos squaredx=3

Sarah Word Problem

She also asked to sate the amplitude , frequency , period , domain , range , phase shift , and reflection
Word problem:
The average monthly temperature of a town can be modelled by the function T(t)= 14.6 sin 0.5( t-1)+9.15. where T is the tempertaure in degrees and t=0 represents january first, t= 1 represents febrauary 1 etc...lkd
a) which month is the average monthly temperature the highest?the lowest?
b) use the model to predict when temp is 0?
c) when is the temperature 20?

T(t)= 14.6 sin 0.5( t-1)+9.15

amp=14.6
freq=.5
Phase shift= .5*-1 = -.5
Period=2pi/.5=4pi or approx 12
Domain= Infinity the sin graph goes on forever on the x-axis
Range= 9.15 then add 14.6 and get a maximum of 23.75 and then 9.15-14.6 and you get minimum of -5.45
reflection= ?

Since you are being asked the month with the maximum temp. You need to say

23.75=14.6 sin 0.5( t-1)+9.15

then you solve for t

subtract 9.15

14.6 =14.6 sin 0.5( t-1)

divide by 14.6

1= sin 0.5(t-1)

sin inverse of 1 = 0.5(t-1)

1.57 = 0.5(t-1)

3.14 = t-1

t=4.14

The maximum temperature is in May as May is equal to 4

When is the temperature 20 or 0

20=14.6 sin 0.5( t-1)+9.15

t= 2.67 or Middle of March

0=14.6 sin 0.5( t-1)+9.15

Solve them the same way as I showed above.