CPhill

1)Let theta be an acute angle such that sin(theta) = 3/5. What is the value of cos(180-theta)?

Note...cos theta  =  4/5

cos (180  - theta)  =

cos (180) cos (theta)   + sin (180) sin (theta)  =

-cos (theta)  +  (0) * sin (theta)  = 

-cos (theta) =

- 4/5

3.The half-life of Radium-226 is 1590 years. If a sample contains 200 mg, how many mg will remain after 1000 years? 

We need to find k....[A  is some original amount]

.5A  = A * e^(k * 1590)   divide  through by A

.5 = e^(k * 1590)      take the Ln  of both sides

Ln (.5)  = Ln e^(k * 1590)  and we can write

Ln (.5)  = (k * 1590)    

k = Ln (.5) / 1590

So.....if the original amount is 200 mg, the amount remaining after 1000 years is

200* e^( Ln (.5) / 1590 * 1000)  ≈  129.3 mg

I couldn't have done it without  your help, EP....LOL!!!

2. A rectangular storage container with an open top is to have a volume of 80 cubic meters. The length of its base is twice the width. Material for the base costs 80 dollars per square meter. Material for the sides costs 9 dollars per square meter. Find the cost of materials for the cheapest such container. 

Minimum cost = 

The dimensions of the base  are   L * W   =  2W * W  = 2W^2

Ans the height  is H

The volume, V, can be expressed as

80  =  L * W * H

80  = 2W * W * H

80 = 2W^2 * H        solving for the height, we have

40/W^2  = H

So....the surface area,SA, can be expressed as

SA  = base area + area of the sides  = 

2W^2 + 2(40/W^2)(2W) + 2(40/W^2)(W)  =

2W^2 + 240/W

So....the cost, C, can be represented  as

C =

2W^2(80)  + (240/W)(9)  = 

160W^2  + 2160/W       take the derivative of this  and set to 0

C '  =  320W - 2160/W^2  = 0

320W^3 - 2160  =  0

W^3  =  2160/320

W^3  =  27/4

W  = 3/∛4 

And the minimum cost is given  by :

160(3/∛4)^2  + 2160 / (3/∛4 )   ≈  $1714.39

1.The consumers will demand 30 units when the price of a product is $53, and 47 units when the price is $33. Find the demand function (express thr price  in terms of the quantity ), assuming it is linear.

p=

We have the points  (30, 53)  and (47, 33)

We need to find the slope....so we have

[ 33 - 53]  / [ 47 - 30 ]  =  [ -20 ] / [ 17] =   -20/17

So we have

p  = (-20/17)(d - 30) + 53

p = (-20/17)d + 600/17 + 53

p = (-20/17)d + 1501/17

500,000  =  A (1.05)^10   divide both sides by (1.05)^10

500,000/(1.05)^10  = A ≈ 

$306,956,63

7000  = 8500(.955)^t      divide both sides by  8500

7000/8500  = .955^t

7000/8500  = .955^t      take the log of both sides

log (7000/8500)  =  log (.955)^t       and we can write

log (7000/8500)  = t * log (.955)  divide both sides by log(.955)

log (7000/8500)  / log (.955)  = t ≈ 4.2  years after 2010  =  2014

y  = x^2  - 3x + 4

x + y  =  4    ⇒   y  =  4 - x

Set the y's equal

x^2 - 3x  + 4   =  4  - x         subtract 4 from both sides

x^2  - 3x    =  - x          add x to both sides

x^2  - 2x  =   0        factor

x ( x - 2)  = 0

So.....setting each factor to 0 and solving, we have

x  = 0                   x  - 2   = 0   add 2 to both sides

                             x  = 2

So....the solutions are   x   = 0    and x  = 2

√a  - 4 = -10   add  4 to both sides

√a  =  -6      there is no real solution to this.....we cannot get a negative from a positive square root

18 = 2√(x - 3)   + 4       subtract 4 from both sides

14  = 2√(x - 3)      divide both sides by 2

7  = √(x - 3)     square both sides

49   = x  - 3     add 3 to both sides

52   = x

√(3x - 4)  = √(5x - 12)   square both sides

3x  - 4  = 5x  - 12      add 12 to both sides, subtract 3x from both sides

8 = 2x     divide both sides by 2

4 = x

Probability of buying 12 cups that don't have any defect  is

88/100 * 87/99 * 86/98 * 85/97* 84/96 * 83/95 * 82/94 * 81/93 * 80/92 * 79/91 * 78/90 * 77/89  ≈

.1955   ≈  19.6%

Probability of buying all 12 of the defective cups  =

12/100 * 11/99 * 10/98 *  9/97 * 8/96 * 7/95 * 6/94 * 5/93 * 4/92 * 3/91 * 2/90 * 1/89  =

           1

__________________      ≈  [ very little chance!!!  ]

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