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We can use the "Ring Method" to solve this

The volume is   =  volume of top function rotated about the x axis from x = .5 to x = 1   less the  volume of the bottom function rotated about the x axis  from x = .5 to x = 1

So  we have

     1                              1

pi  ∫   (x + 1/x)^2 dx    -    pi  ∫  ( 2x)^2 dx    =

     .5                            .5

     1                                          1

pi  ∫   x^2 + 2 + 1 /x^2  dx  -  pi  ∫  4x^2   dx    =

    .5                                         .5

     1

pi  ∫   2  + 1/x^2  - 3x^2   dx  =

    5

             1              1             1

pi (    2x ]      - 1 /x ]     -  x^3 ]    )   =

            .5              .5            .5

pi * ( [ 2 - 1 ] - [ 1 - 2 ] - [ 1 - .125 ]  )  =

pi  ( 1 +  1  - 0.875  )  =

[  1.125 pi   ]  units^3

Let n be the first number and 4n the second

N + 4N  = 50

5N = 50    divide both sides by 5

N = 10

And the second number = 4 N =  4 (10)  =  40

OK..if January means that x =  0.....then March means that x = 2

So we have

f(2) =  12.2 cos ( pi * 2 / 6) + 54.9  =    12.2cos ( pi /3) + 54.9  =   12.2 * [ 1/2] + 54.9 = 

6.1 + 54.9  =

61   (degrees)

So sorry for the late reply! Thank you for your help!!

Note that we can use the Pythagorean Theorem to find the diagonasl of a square

diagonal  =  sqrt (side^2 + side^2)  =  sqrt ( 2 * side^2)  =  side * sqrt (2)

So....the side =   diagonal /  sqrt (2)

So we have that the side =  10 / sqrt (2)

Multiply top/bottom by sqrt (2)   and we get that

side  =  10 sqrt (2) / [ sqrt (2) * sqrt (2) ]  =  10 sqrt (2) / 2   =  5 sqrt (2)

At   0 sec.....the painted spot is at its lowest point

The period (one complete revolution) is 1.5 sec....it turns through 360°  during this time....

So.....(1/2) of this = .75sec.....so.....it turns through 180°  in .75 sec

Where did you get the radical 2 from?

f(March) = 12.2 cos (pi6 2) + 54.9       ( Jan = 0    feb = 1   mar = 2)  =    I'll let you crunch the numbers.  (are you SURE it isn't pi/6 *  x  ??)

(make sure you use radians....not degrees)

The side of the square =    10 / √2   cm   =  5√2 cm

And the perimeter is  4 times this  =  20√2 cm

Her gross pay was $ 34854

For 2012    SS TAX WAS 6.2%    Item 4 should then be   .062 * 34854 = 2160.95

For 2012    Medicare tax was 1.45%    Item 6 should be     .0145 * 34854 = 505.38

FICA tax is Social security + medicare taxes

She paid  $1762.39 state income tax

From table, with $32104 taxable as head of houshold she owes $4199 in fed tax ....she paid 4182.48 ...she owes the difference $16.52

At  t = 0.5 seconds.....the pendulum is at its resting position

At  1 sec ...it is at its max displacement to the right

At  1.5 sec...it returns to its resting position

So ...  1.5 - 0. 5 =      1 second is required

(1)    Reflect  the function  about the y axis

(2)    Rotate the graph of (1)     90°  clock-wise

This will be the inverse 

First one  :  https://www.desmos.com/calculator/f34fdy3pga

Second one :  https://www.desmos.com/calculator/i1l4bgsjmp

Third one :   https://www.desmos.com/calculator/exbmqsekow

Last one : https://www.desmos.com/calculator/pqzzmgj0gg

\(-7 < 7 \tan(x) < 7\\ -1 < \tan(x) < 1\)

\(\text{the easiest way to find }x \text{ is to look at at he graph of }\tan{x} \)

\(\text{we see the inequality is satisfied for }\\ x \in \left[0,\dfrac \pi 4\right) \cup \left(\dfrac{3\pi}{4},\dfrac{5\pi}{4}\right)\cup \left(\dfrac{7\pi}{4},2\pi\right]\)

Thank you!

First column                   Second column                 Third column

9.2 X 10^(-4)                  2.1 X 10^(-4)                       3.4 X 10^(-5)                   

2.1 X 10^(-3)                  8.3 X 10^(-5)                       2.8 X 10^(-7)

                                                                                   7.2 X 10^(-5)

Thank you so much for your help!! I appreciate it.

The inverse just reverses the coordinates...so we have

(-3, 1), ( 3 , -2), (1, 5) , (4, 6)

We don't have any single "x"  associated with two different "y's".....so.....the inverse is a function

(7, -5), ( -8, -6) , ( -2, 1), ( 3, 10 )

Again...no single x is  associated with two different "y's"....so....the inverse relation is a function

Here's the way we can test for an inverse.....put first function into the second....then, put the second function into the first.....if we get an "x" as a result in both cases....they are inverses

First one

(x - 6) + 6   =   x

(x + 6) - 6  =   x         these are inverses

Second one

2  [ ( x + 7) / 2 ] - 7   =    x + 7 - 7   =  x

 ( [ 2x - 7 ] + 7 )  / 2  =      [2x] / 2   =  x             these are inverses

We  can use the Pythagorean Theorem here :

UM =  sqrt (JM^2 - JU^2 )   =   sqrt ( 12^2 - 8^2)  =  sqrt ( 144 - 64)  = sqrt ( 80)  =  4sqrt(5) units

FV = PV (1+ i)^n      i = 2.1 % per year divided by 12 months = .00175 monthly       n = 12 months * 16 yrs = 192 periods

     = 76000 (1.00175)^192 = $ 106318.54

Same equation   PV = 81000      i = periodic interest = 2.2% / 4   (quarterly ) = .0055        n = periods = 19 yrs x 4/yr = 76  periods

                                 81000(1.0055)^76 = $ 122892   (rounded to nearest dollar)

Let the price of the small size = P   and let the quantity contained = Q

So....the unit price is     P /Q

Let the price of the meduim size =  1.5P

And let the quantity contained be   (2 - ..20)Q  =  1.8Q

So.....the unit price is    (1.5)P /  (1.8)Q   =  (5/6)P/Q

The large size price is (1.5)(1.3)P = 1.95 P

And the quantity is 2Q

So.....the unit price is  (1/95  P) / (  2Q)   =  (19/20)P/Q

So.....the best value from best to worst is

 ( 5/6)P/Q     (19/20)P/Q          P /Q  =

   M                  L                      S

Here you are :

The tail (caboose as they used to have)  starts 1 mile from the mouth of the tunnel....ti must travel THAT mile PLUS the length of the tunnel in 3 minutes

t = tunnel length

t + 1  = distance the caboose must travel in 3 minutes     3 minutes = 1/20 hr

(t+1) mile  /  60 m/hr  = 1/20 hr

t + 1 = 60/20

t = 2 miles long