geno3141

y  = (3x + 15)/4          x² + y²  =  36

Substituting:  x² + [ (3x + 15)/4 ]²  =  36

          x² + (9x² + 45x + 225) / 16  =  36

               16x² + 9x² + 45x + 225  =  576

                          25x² + 45x - 351  =  0

Using the quadratic formula:     x₁  =  [-9 + 12sqrt(3) ] / 5          or          x₂  =  [-9 - 12sqrt(3) ] / 5 

Substituting  x₁  into  y  = (3x + 15)/4  results in  y₁  =  [12 + 9sqrt(3) ] / 5

Substituting  x2  into  y  = (3x + 15)/4  results in  y2  =  [12 - 9sqrt(3) ] / 5

A  =  ( [-9 + 12sqrt(3) ] / 5 ,  [12 + 9sqrt(3) ] / 5 )

B  =  ( [-9 - 12sqrt(3) ] / 5  ,  [12 - 9sqrt(3) ] / 5 )

Now, to find the length of AB, place the values of  (x₁, y₁)  and  (x₂, y₂)  into the distance formula:

Distance  =  sqrt[ (x₂ - x₁)² + (y₂ - y₁)² ] 

Yeah --I messed up -- I used the formula for the area of a circle instead of the formula for the circumference of the circle:

Correcting:

  Circumference  =  2·pi·radius

  (1/8)·2·pi·6  =  answer #1

  (1/8)·2·pi·3  =  answer #2

Now, subtract answer #2 from answer #1.

Sorry ...

y varies inversely as sqrt(x)     --->     y = k / sqrt(x)

x = 2, y = 4                              --->      4  =  k / sqrt(2)     --->     4·sqrt(2)  =  k

x = ?, y = 1                              --->     1  =  4·sqrt(2) / sqrt(x)   

                                     --->     1·sqrt(x)  =  4·sqrt(2)

                                     --->         sqrt(x)  =  4·sqrt(2)

                                     --->    [ sqrt(x) ]²  =  [ 4·sqrt(2) ]²

                                     --->                  x  =  32

If you are supposed to simplify by using the distributive property:

      (p - 2q - 4)3  =  p·3 - 2q·3 - 4·3  =  3p - 6q - 12

My plan is to find the distance between the centers of the two circles and subtract the lengths of the radii of the two circles.

(x - 9)² + (y - 5)²  =  6.25     --->     C(9,5)     r = 2.5

(x + 6)² + (y + 3)²  =  49      --->     C(-6,-3)     r = 7

Distance between (9,5) and (-6,-3)  =  sqrt[ (-6 - 9)² + (-3 - 5)² ]  =  sqrt[ (-15)² + (-8)² ] 

                                                         =  sqrt( 225 + 64)  =  sqrt(289)  =  17

Distacne between the two circles:  17 - 2.5 - 7  =  7.5

Point-slope formula:  y - y₁  =  m(x - x₁)     

Since we don't know the slope, we have it find the slope first:

  m  =  [ y₂ - y₁ ] / [ x₂ - x₁ ]   =   [0 - 3] / [1 - -3]  =  -3/4

  y - y₁  =  m(x - x₁)     --->     y - 3  =  (-3/4)(x - -3)     --->     y - 3  =  (-3/4)(x + 3)

                               multiply both sides by 4:                     4y - 12  =  -3(x + 3)

                               simplify:                                               4y - 12  =  -3x - 9

                                                                                                   4y  =  -3x + 3

                               divide by 4:                                                   y  =  (-3/4)x + 3/4

This is the difference in length of a 45^o arc of a circle with a radius of 6" and the length of a 45^o arc of a circle with a radius of 3".

Use the formula for the circumference of a circle:  C  =  2·pi·radius

Since 45^o is 1/8^th of a complete circle:

-- the length of a 45^o arc of a circle with a radius of 6"  =  (1/8)·2·pi·6  =  answer #1

-- the length of a 45^o arc of a circle with a radius of 3"  =  (1/8)·2·pi·3 =  answer #2

Subtract answer #2 from answer #1

I don't know if this is the easiest way, but it's the way that I saw.

I'm going to find the area of the rhombus and then use the formula:  Area  =  ½ · d₁ · d₂ 

Since a rhombus has 4 equal sides, each side has value:  68 / 4  =  17.

The area of the triangle formed by two sides of the rhombus and the diagonal can be found by using Heron's

formula:  Area  =  sqrt[ s(s - a)(s - b)(s - c) ]  where a, b, and c are the lengths of the three sides and 

s is the semiperimeter:  s  =  (a + b + c) / 2.

s  =  (17 + 17 + 16) / 2  =  25

Area(triangle)  =  sqrt[ 25(25 - 17)(25 - 17)(25 - 16) ]  =  120

Therefore, the area of the rhombus  =  2 x 120  =  240

Now using the formula:  Area  =  ½ · d₁ · d₂    --->   240  =  ½ · 16 · d₂   --->   d₂  =  30

The consecutive integers are:  x,  x +1,  x + 2,  x + 3, x + 4

The mean of these integers is:  x + 2

Since the mean is 21:  x + 2  =  21

which means that x (which is the smallest) is 19.

If you don't get at least one head, you must get all tails.

To find the probability of getting at least one head, subtract the probability of getting all tails from 1.000.

One toss:         1.000 - all heads  = 1.000 - 0.80  =  0.20

Two tosses:     1.000 - all heads  = 1.000 - 0.80x0.80  =  0.36

Three tosses:  1.000 - all heads  = 1.000 - 0.80x0.80x 0.80  =  0.488

Four tosses:    1.000 - all heads  = 1.000 - 0.80x0.80x.80x0.80  =  0.5904

It takes four tosses.