CPhill

To find these points..... first  find  the x  intersection of the graphs of

y  = (5/12)x  +  2     and    x^2  +  (y -2)^2  = 9

Manipulating the first....we have that  y  -  2  =  (5/12)x

Subbing this into the second, we have that

x^2  +  (5/12* x)^2  =  9      simplify

x^2  +  25x^2/ 144  = 9 

x^2  +  (25/144)x^2  = 9

169/144x^2  =  9

x^2  =  9*144/ 169       take the positive and negative roots

x  = ±√  [ 9 * 144 / 169 ]

x =  ± 36/13

So..using   y   =  (5/12)x  + 2...the y intersection points are

y  =  (5/12)(36/13)  +  2  =  (180/156) + 2  =  41/13

And 

y  =   (5/12)(-36/13)  +  2  =  (-180/156)  + 2  =  11/13

So....the points of intersection are

(36/13, 41/13)   and  (  -36/13, 11/13)

So  using these   points and the slope  (-12/5)   we have the  equations

y  =  (-12/5)( x - 36/13)  +41/13

So...when   x  = 0, the y intercept of this line  is  (-12/5)(-36/13)  + 41/13  = 49/5  = 9.8

And

y  =  (-12/5)( x +  36/13)  +  11/13

And.....when x  = 0, the y intercept of this line  is (-12/5)(36/13)  + 11/13  = -29/5  = -5.8 

Here's  a graph : 

https://www.desmos.com/calculator/ctru5i68ra

The other vertex  of the rectangle is  the circle's center  =   (5, -2)

Look  at the image here :  

The region common to both areas is just  the quarter circle  area  of the given circle

And this circle has a radius  of 3.....so....the quarter circle has an area  of

pi (r^2)  / 4  =   pi (3^2)  / 4  =    (9/4)pi   units ^2

x^2  +  y^2  ≤  36   (1)

y  =  x  - 4     (2)

Look at the graph, here  : https://www.desmos.com/calculator/leyhcfzzep

The integer pairs that make this true  are  :

(-1, -5)   (0, -4)  (1, -3)  (2, -2)  ( 3, -1)   ( 4, 0)   ( 5, 1)

\($\dfrac{30+19i}{4+9i}$\)       multiply top/bottm by  4  - 9i

[ 30  +  19i ] [ 4 - 9i ]  / (  4 + 9i) (4 - 9i) ]

[ 120 + 76i  - 270i   -  171i^2 ]  /  [ 16 -  81i^2  ]

[  120  +  171  -  194i ]  /  [ 16 + 81  ]

[ 291  -  194i  ] / [ 97 ]

(291/ 97)  -  (194/97) i

3 - 2i

\(\frac{1-i}{2+3i}\)     Multiply  top/bottom by   2 - 3i

[ 1  - i  ] [ 2 - 3i ]  /  [ ( 2 + 3i) ( 2 - 3i)  ]  

[ 2 - 2i - 3i + 3i^2  ] /  [ 4  - 9i^2 ]

[ 2  -  3  - 2i -  3i  ]  / [  4 + 9 ]

[ -1  - 5i  ] / [ 13 ] 

log₃ 2187

We can use  the change of base theorem, here

The answer is given by

log 2187  / log 3  =    [  the onsite calculator gives  6.99999.....]  

The actual answer is 7

To see this....we are asking....What exponent do we put on 3 to get 2187 ??

So

3⁷  = 2187

Answered here :

http://web2.0calc.com/questions/help_90006#r1

I think I answered the second one here :

http://web2.0calc.com/questions/help_49408#r1

First one

12x^2  - 64  =  -x^4     add  x^4  o both sides

x^4  +  12x^2  -  64  =  0     factor as

(x^2  + 16)  (x^2  - 4)  = 0

(x^2  + 16)  (x + 2) (x - 2)  =  0

We only have 4 solutions.....  setting the last two factors to 0 and solving for x  we get that x  = 2   and x  = -2

So....the first answer is correct

7%  increase  per year  =  (1.07)...so.....

f(x)  =  295  (1.07)^x

After 2 years

f (x)  =  295 (1/07)²    =  338

y  =  487 (1/2)^{(1/36 * x)}

After  x  =   5 days

y  =  487  (1/2^{)(1/36 * 5})  ≈ 442.302  kg