geno3141

I don't understand --

-- if line m has equatioin  y = 2x - 5

-- if line n has equation    y = x + 22

they won't be parallel because they don't have the same slope (the slope of m is 2 while the slope of n is 1).

Do you mean that they aren't parallel and you need to know where they intersect?

cos(a - b)  = cos(a)·cos(b) + sin(a)·sin(b)

Since tan(a)  =  -12/5  and is in quadrant II, sin(a) = 12/13  and  cos(a) = -5/13

  (It's a 5-12-13 right triangle, with x-value negative and y-value positive.)

Since cos(b)  =  3/5  and is in quadrant IV, sin(b)  =  -4/5.

  (It's a 3-4-5 right triangle, with x-value positive and y-value negative.)

Place these values into the formula to get your answer.

Yes, this is correct.

The linear velocities must be the same because they are connected by the chain. The same amount of chain must be traveled by each gear. 

The angular velocities aren't the same, for the smaller gear must travel faster to "keep up", so it must have a faster angular velocity.

The answer of 648p cm per minute is correct because:

-- the smaller gear has a radius of 3 cm, which gives it a circumference of 6 pi cm.

-- therefore, in each revolution, it moves 6 pi cm in a linear distance.

-- since it completes 36 revolutions, it completes 36 x 6 pi cm  =  216 pi cm

-- since it completes 216 pi cm in 20 seconds, it will complete 216 pi x 3  = 648 pi cm in one minute.

1.6 minutes is 1.6 x 60 = 96 seconds.

Its angular velocity is 6.2 radians per second:  6.2 radians/second x 96 seconds = 595.2 radians.

One revolution of the fan is one circumference of the fan, which is 2pi radians.

Dividing 595.2 radians by 2pi radians gives us an answer of 94.7 revolutions.

9 ft is 9 x 12 = 108 in

12 ft is 12 x 12 = 144 in.

The area of the floor is 108 in x 144 in = 15,552 sq in.

Each tile is 4 in x 6 in = 24 sq in.

Dividing  15,542 sq in by 24 sq in gives an answer of 648 tiles.

However, we should check to see if the answer is possible.

Is it possible to have the tiles arranged so that the 4 in side of the tiles fits the 108 in length of the room and the 6 in side of the tiles fits the 144 in length of the room?

   108 in divided by 4 in gives us 27 tiles

   144 in divided by 6 in gives us 24 tiles

   27 tiles in one direction and 24 tiles in the other direction gives us 27 x 24 = 648 tiles.

Also, it is possible to have the tiles arranged so the the 6 in side of the tiles fits the 108 in length of the room and the 4 in side of the tiles fits the 144 in length of the room?

   108 in divided by 6 in gives us 18 tiles

   144 in divided by 4 in gives us 36 tiles.

   18 tiles in one direction and 36 tiles in the other direction gives us 18 x 36 = 648 tiles.

So, we can fit them either way.

Let's find where   y  =  3x² - 14x   and   x  =  y   intersect by replace the y-term in the first equation with x:

      x  =  3x² - 14x      --->     0  =  3x² - 15x    --->   0  =  3x(x - 5)   --->   x = 0   or   x = 5

The value of  a  is  5.

Long way:  find the slope of the original line and use the point-slope form:

    Original line:  5x - 2y  =  6 

         solve for y:     -2y  =  -5x + 6

                                   y  =  (5/2)x - 3

         slope  =  5/2

    Use point-slope form to find the equation of the new line: 

                   y - y₁  =  m(x - x₁)

        --->     y - -4  =  (5/2)·(x - -2)

                    y + 4  =  (5/2)(x + 2)

                    y + 4  =  (5/2)x + 5

                         y  =  (5/2)x + 1

       Rewriting:  2y =  5x + 2                             (multiplied both sides by 2)

               -5x + 2y  =  2

                 5x - 2y  =  -2

Short way:     notice that the old equation   5x - 2y  =  6   and the new equation   5x - 2y  =  -2

have the same coefficients of the x-term and the y-term and their only difference is the constant.

So, to find the new equation, start with  5x - 2y  =

and put in the point (-2, -4) and solve:   5(-2) - 2(-4)  =  -10 + 8  =  -2

This will give us the new constant, so the new equation is  5x - 2y  =  -2  

Number of eclairs      =  3

Number of humbugs  =  2x + 6

Number of mints        =  x

Total number             =  3 + 2x + 6 + x  =  3x + 9

Probability of being an eclair  =  3 / (3x + 9)

Probability of being an eclair  =  1/15

Setting these two equal to each other:  3/(3x + 9)  =  1/15

                     cross multiply:                               45  =  3x + 9

                     subtracting:                                   36  =  3x

                     dividing:                                         12  =  x

Number of humbugs  =  2(12) + 6  =  30

Number of mints        =  12

Total number              =  45

Probability of being a humbug  =  30 / 45

In parallelogram ABCD,  AB = CD  and  AB || CD.

In parallelogram CDFE,  DC = FE  and  DC || FE.

Therefore,  AB = FE  and  AB || EF.

Since a pair of sides of quadrilater ABEF are both parallel and congruent, ABEF is a parallelogram.

For machne A:  with an input of  x  its output will be 5x - 4

For machine B: with an input of  x  its output will be 3x + 8

If machine A's output is three times machine B's output:  5x - 4  =  3(3x + 8)

                                                                                           5x - 4  =  9x + 24

               subtract 5x from both sides:                                      -4  =  4x + 24

               subtract 24 from both sides:                                    -28  =  4x

               divide both sides by 4:                                               -7  =  x

Checking:  machine A:  5x - 4  =  5(-7) - 4  =  -35 - 4  =  -39

                  machine B:  3x + 8  =  3(-7) + 8  =  -21 + 8  =  -13

And:  -39 is three times -13.