CPhill

Here is a YouTube video explaining the procedure for   (1)

https://www.youtube.com/watch?v=09aYOtI7v2g

And here's another one for  (2)

https://www.youtube.com/watch?v=YEX2mhvbYuo

3) We have to know the measure of θ   before we can provide a definite answer

But....the  area is given by

(1/2)* radius^2 * θ        if  θ  is in radians

or

pi * radius^2  *   (θ / 360)     if  θ  is in degrees

A)  Allowing for repeated letters/numbers  we have   (26)^2 * (10)^2  =  67,600  possible codes

So....the chance of guessing correctly the first time is    1 /67,600 ≈ 0.0000148  =  .00148%

B)  With M being  the first letter, there are  26 * 10^2  codes  =  2600 possible codes

The probability of a correct guess  on the first attempt  =  1/2600  ≈ 0.0003846  = .03846 %

A)  Makes the basket, misses the basket, Makes the basket

(.80) (.20) (.80)  =  (.80)^2 (.20)  = 0.128  =  12.8 %

B) Makes all 3 shots

(.80)^3    =  0.512  =  51.2 %

C) Misses all 3 shots

(.20)^3  =  0.008  =  .8 %

D) Misses the frist two and makes the third

(.20)^2 (.80)  =  0.032  =  3.2 %

4. find the arc length of the curve from t=0 to t=2 whose derivatives in paramertic form are dx/dt = cos(t) and dy/dt=ln(t+1). Write answer to two decimal places. Not sure about this one!!

2

 ∫   √    ( dx/dt)^2  +  (dy/dt)^2  )   dt

0

2

 ∫   √  [ cos^2(t)  +  (ln (t + 1])^2  ] dt      ≈  1.91 

0

1. what is the length of the curve with parateric equtaions x=1-2cos(t), y=2-sin(t) from t=0 to tot=pi?

dx/dt = 2sin(t)    ....  [ dx/dt]^2  =  4sin^2(t)

dy/dt  = -cos(t).....    [dy/dt]^2  =  cos^2(t)

The arc length  is  found as

pi

∫   √ (  [ dx/dt]^2  + [ dy/dt]^2 ) dt   =

0

pi

∫   √ (  4sin^2(t)  + cos^2 (t) ) dt

0

pi

∫   √ (  4sin^2(t)  + 1 - sin^2(t) ) dt

0

pi

∫   √ (  3sin^2(t)  + 1 ) dt   ≈         4.84422

0

What is the length of the path along the graph of y=sqrt(9-x^2) between x=0 and x=2?

dy/dx  =   (1/2) (9 - x^2)^(-1/2) ( -2x)  =  -x / ( 9 - x^2)^(1/2)

[dy/dx]^2  =  x^2 / [ 9 - x^2]

The arc length  is given by

2

∫      √ ( 1 + [dy/dx]^2  )  dx   =  

0

2         

∫   √ ( 1 +    x^2 / [ 9 - x^2]  ) dx   =

0

2

∫   √ ( 9  / [ 9 - x^2]  ) dx  =

0

    2

3  ∫   √ ( 1 / [ 9 - x^2]  ) dx  =

   0

let  x  =  3sin (θ)   →    x^2    =  9sin^2( θ)      

dx  =  3cos (θ) dθ

Change the limits of integration

when x  = 2               when x  =  0

2/3  = sin (θ)              0  = sin (θ)

θ = arcsin(2/3)           θ =  0

So  we have

   arcsin(2/3)

3   ∫  √ ( 1 / [ 9  -  9sin^2( θ) ] ) 3 cos ( θ) dθ   =

    0

 arcsin(2/3)

3   ∫ (1/3)  √ ( 1 / [ 1  -  sin^2( θ) ] )  3 cos ( θ) dθ   =

    0

 arcsin(2/3)

3   ∫  √ ( 1 / [ 1  -  sin^2( θ) ] ) cos ( θ) dθ   =

    0

 arcsin(2/3)

3   ∫   √ ( 1 / [cos^2 ( θ) ]  ) cos ( θ) dθ   =

    0

 arcsin(2/3)

3   ∫    ( 1 / [cos ( θ) ] ) cos ( θ) dθ   =

    0

 arcsin(2/3)

3   ∫    dθ   =

    0

          arcsin(2/3)

3 [ θ ]                       =     3 [  arcsin(2/3)  -  0 ]   =    3  arcsin(2/3)

             0

Answer "b"

 P(x) = -.05x + 53,708

The population in 2005  was 53,708   [maybe the first statement is a typo ?? ]

We have that

Volume_sphere  =  (4/3)pi (radius)^3

9000  =  (4/3)pi (radius)^3     divide both sides by (4/3)pi

9000/ [ (4/3)pi ]  =  radius^3       takethe cube root of both sides

 ∛  ( 9000/ [(4/3)pi] )  =  radius  ≈ 12.9  units

And the diameter is twice this ≈  25.8  units

x^2 + 10x + y^2 + 12y  + 52  = 0     subtract 52 from both sides

x^2 + 10x + y^2 + 12y   =  -52

Complete the square on x.....take (1/2) of 10  = 5....square it = 25,  add it to both sides

x^2 + 10x  +  25 + y ^2  + 12y =  -52 + 25

Complete the square on y....take (1/2) of 12  = 6....square it  = 36, add it to both sides

(x^2 + 10x + 25) +(  y^2 + 12y + 36)  =   -52 + 25 + 36

Factor the stuff in parentheses- both will be perfect square trinomials   ....simplify the right side

(x + 5)^2  +  ( y + 6)^2  =  9 

7