geno3141

Currently, the ratio of milk to water is 5:3. This means that 5/8^ths of the mixture is milk.

If you pour out 4 liters from 20 liters, you now have 16 liters.

If 5/8^ths of this mixture is milk, (5/8) x 16 liters is 10 liters of milk.

To these 10 liters of milk and 6 liters of water, you add 4 liters of milk.

You now have a mixture of 14 liters of milk and 6 liters of water.

This gives a ratio of 14:6 or 7:3 of milk to water.

The formula for each corner angle of a regular polygon is  (n - 2)(180^o) / n

There are 360^o around a point.

Each corner angle of a square has 90^o.

Each corner angle of a regular pentagon has (5 - 2)(180^o) / 5  =  108^o.

Subtracting:  360^o - 90^o - 108^o  =  162^o.

Since each corner angle of this regular polygon has 108^o, we can find the number of sides:

                    (n - 2)(180^o​)/n  =  162^o

                       (n - 2)(180^o​)  =  162^o · n

                     180^o·n - 360^o  =  162^o·n

                  180^o·n - 162^o·n  = 360^o  

                                  18^o·n  =  360^o

                                         n  =  20 sides

Joe's rate:      1 garage every 24 hours  =  1/24

Bobby's rate:  1 garage every 12 hours  =  1/12

Father's rate:  1 garage every 4 hours    =  1/4

time:  t

(rate)·(time)  +  (rate)·(time)  +  (rate)·(time)  =  1 job completed

                         (1/24)·t  +  (1/12)·t  + (1/4)·t  =  1

                                                      t + 2t + 6t  =  24                  (multiply by 24)

                                                                  9t  =  24

                                                                    t  =  24/9

                                                                    t  =  2 2/3 hrs  =  2 hrs 40 mins.

A formula for finding the area of a rhombus, given the two diagonals d₁ and d₂, is:  Area  =  ½·d₁·d₂

Since the diagonals of a rhombus are perpendicular, they form four right triangles.

Since the sum of its two diagonals is 34, the sum of one-half of one diagonal (one side of a right triangle)

and one-half of the other diagonal (the other side of a right triangle) is 17.

Let one side be  x  then the other side is  17 - x.

Using the Pythagorean Theorem:  (x)² + (17 - x)²  =  13²

                                                x² + 289 - 34x + x²  =  169

                                                      2x² - 34x + 120  =  0

                                                          x² - 17x + 60  =  0

                                                          (x - 5)(x - 12)  =  0

                                                                      x = 5  or  x = 12

So, the diagonals are  10  and  24  and the area is  ½·10·24  =  120

Write  0.630 630 630 ... as a fraction.

Let            x  =  0.630 630 630 ...

then  1000x  =  630.630 630 630 ...

Subtracting:   1000x  =  630.630 630 630 ...

                          -   x  =      0.630 630 630 ...

                        999x  =  630

dividing:                 x  =  630 / 999

simplifying:            x  =  70/111

There are  20 x 19 x 18  difference possibilities  =  6840

How many of these possibilities have both Smith and Jones?

There could be:  S, J, x  or  S, x, J  or  x, S, J  or  J, S, x  or  J, x, S  or  x, J, S.

where  x  could be any of the other 18 persons, for a total of 6 x 18  =  108  possibilities.

Subtracting these, we get  6840 - 108  =  6732  that don't have both Smith and Jones.

x = r·cos(theta)  =  sqrt(30)·cos(110^o)  =  -1.873

y = r·sin(theta)  =  sqrt(30)·sin(110^o)  =  5.147

The full arc of the circle is 2·pi·25 cm  =  50·pi cm.

The arc is only 5·pi/12.

The portion of the circle is:  ( 5·pi/12 ) / ( 100·pi )  =  1 / 120

This angle represents 1/120^th o the circle which is (1/120)·2 pi  =  pi/60

To make the typing simpler, I'm going to substitue x for sqrt(5).

Numerator  =  sqrt(x + 2) + sqrt(x - 2)

Denominator  =  sqrt(x + 1)

I'm going to square this fraction, reduce it, and then find the square root to get the answer.

Numerator:  [ sqrt(x + 2) + sqrt(x - 2) ]²  =  [ (x + 2) + 2·sqrt(x + 2)·sqrt(x - 2) + (x - 2) ]

       =  [ 2x + 2·sqrt(x + 2)·sqrt(x - 2) ]  

       =  [ 2x + 2·sqrt( x² - 4 ) ]

But, since x = sqrt(5), x² = 5:

       =  [ 2x + 2·sqrt( 5 - 4 ) ]

       =  [ 2x + 2·1 ]

       =  [ 2x + 2 ]

       =  2( x + 1 )

Denominator:  [ sqrt(x + 1) ]²  =  x + 1

Combining:  2( x + 1 ) / ( x + 1 )  =  2

Since we squared this fraction, we'll need to find the square root of this to get our answer:  sqrt(2)