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is 1/5*30=6?

Hello Guest!

Yes, \(1/5*30= 1:5\cdot 30=6\).

The multiplication and division take place in succession from left to right.

  !

Yes but what are the combonations?

Perhaps this graph will help you answer the questions:

If the roots are p, q and r then:

(x-p)(x-q)(x-r) = x^3 - (p+q+r)x^2 + (pq+pr+qr)x - pqr

Compare with the x^2 term of the original cubic to see that p+q+r = 7

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Miko and Freya saved the same amount of money. After Miko lent $32 to Freya, Freya had 5 times as much money as Miko. How much did Freya have in the end?

Miko und Freya haben den gleichen Geldbetrag gespart. Nachdem Miko Freya 32 Dollar geliehen hatte, hatte Freya fünfmal so viel Geld wie Miko. Wie viel hatte Freya am Ende?

Hello Guest!

The amount of money saved is x.

\(5(x-32)=x+32\\ 5x-160=x+32\\ 4x=192\)

\(x=48\)

\(Freya\ have\ in\ the\ end\ $48+$32=$80\)

  !

Hi Melody: I don't see any "bugs" in the code !. C=20. [20^3 - 20 =7980 mod 12 =0]......etc. Here are the results of all the 60 numbers that I have listed. Here is another short code taking them to "mod 12" and they all give "zero", which means they are ALL multiples of 12.

listforeach(n, d=(7980 , 9240 , 12144 , 13800 , 15600 , 19656 , 21924 , 24360 , 29760 , 32736 , 35904 , 42840 , 46620 , 50616 , 59280 , 63960 , 68880 , 79464 , 85140 , 91080 , 103776 , 110544 , 117600 , 132600 , 140556 , 148824 , 166320 , 175560 , 185136 , 205320 , 215940 , 226920 , 249984 , 262080 , 274560 , 300696 , 314364 , 328440 , 357840 , 373176 , 388944 , 421800 , 438900 , 456456 , 492960 , 511920 , 531360 , 571704 , 592620 , 614040 , 658416 , 681384 , 704880 , 753480 , 778596 , 804264 , 857280 , 884640 , 912576 , 970200), c=n%12) 

OUTPUT=(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)

When 125 is added to a number, it becomes a perfect square. When 176 is added to that same number, it also becomes a perfect square. What is the number?

500 +125 = 625       sqrt(625) = 25

500 + 176 = 676      sqrt(676) = 26  

Sorry Guest, I had contradicted you but I need to look again.   

The only way that a  number is divisible  by  5 is if it ends in  0  or  5

So..........we  must have  the form

___   ____    ____    5

We  have  6  other digits  to  choose from....so.....the total possible  4 digit numbers  divisible by 5  with no digits repeated  =

6 choices   * 5 choices  * 4 choices   =

120

Leo needs to drive from Boston to Chicago. He planned to drive for 20 hours at an average speed of 50 mph. But, he drove at an average speed of 60 mph for the first 5 hours. During the remaining part of the trip, his average speed was 50 mph. How much earlier did he arrive in Chicago than he usually planned

Well..... using the original calculation.......the total   distance  must  be  20 * 50     = 1000 miles

So   he traveled  60 * 5   = 300 miles   and the  distance he  had left to travel  = 1000 - 300 = 700 miles

So

Distance  / Rate  = Time

So

700 / 50   =    14   hrs

So....his total travel  time =   5  + 14   = 19  hrs

So.........he arrived one hour earlier than he originally planned

We  need  the slant height  of the faces  

This will be  =  sqrt  [ 1/2 of side  length ^2   +  height ^2 ]  =  [ 10^2  + 17^2 ]  =  sqrt (389) m

The  area of the  four faces =

4  [ area of  one face ]  =

4  [  area of a triangle with a base of 20  and height of sqrt (389) ]   =

4 (1/2) base * height  =

2  (20) sqrt (389) 

40 sqrt (389) ≈   

788.9 m^2

Assuming  the  the state fee is on the total amount (after tax amount) 

131.30 (1 + .065) (1.30)  = $181.78

Assuming that  the  30%  fee is  only on the original  amount  (before tax amount )

131.30 (1+.065)  + 131.30 (.30)  =

131.30  ( 1 + .065 + .30)  = $179.22

Let  x  be  the part  of the wire reserved for the  circle

This  must  be  the perimeter of the  circle

Perimeter =  2pi*r

x  =2pi* r

[ x / (2pi) ]   = r

So   the  area  of the  circle =  pi* r^2 =  pi  [ x / (2pi)]^2 = [x^2] / [ 4pi ]

The part  reserved for the squre is just the perimeter of the square  =   [ 10- x ]

The side of the square  =  perimeter /4 =   [ 10 - x ] /4

Area of the square =  side^2  =  [ 10 - x ]^2  / 16

So....the  total area of the circle and square , A, can be expressed  as

A =  x^2 / [ 4pi ]  +  [ 10 - x ]^2 / 16         simplify

A = x^2 / [ 4pi ]  +  [ x^2 - 20x + 100] /16

A =  x^2 / [4pi]  + (1/16)x^2  - (5/4)x  + 25/4

This  is  most easily done with Calculus.....I'll  graph the  curve to show you that  the  value for x obtained is the  correct one

Take the  derivative  of  A  and set  it to  0 

2x /(4pi) + (2/16)x  - 5/4  = 0

x / [ 2pi]  +  x /8  = 5/4

x  [   1/[ 2pi] +  1/8]  =  5/4 

x [ 1/(2pi) + 1/8 ]  = 1.25   

x  =  1.25  / [  1/(2pi)  + 1/8]  ≈  4.399  ≈  4.4  

Cut the wire  at  about  4.4 inches.....this is the part reserved  for the  circle

So  the part reserved for the square  =10 - 4.4  = 5.6  in

Here is a graph  showing that the  value for  x  is  correct  :

https://www.desmos.com/calculator/d8t5qos9c3

P       R          Q

               324      

Note  that if it took  Jack 9 hrs  to go from R  to Q....then the distance  from Q to R must  be  9 * 36  = 324 miles

So.....let T  be  the time it takes both to reach  R

So....Jills Rate  (in mph)  must   be  : 

[Distance from Q ot R] / [Time to  travel from Q to R] =   324 / T

So....the  distance that Jack travels  must  be  36T  + 324

And  the distance that Jill travels  must be  324  +  her rate *  time to  travel  from R to P  =

 324  + (324/ T)  *  4

Since  these distances are the same, we have this equation :

36T  + 324   =  324  + (324/ T)  *  4          subtract   324  from both sides

36T =  ( 324/T ) * 4

36T  = 1296 /T      multiply both sides  by  T

36T^2   =1296       divide both sides by  36

T^2  =36    take the square root of both sides

T = 6  hrs

So  Jill's rate must be      324 /  6   = 54 mph

If I have 50 20c stamps, how much dollars do I have altogether?   

You can't tell how many dollars you have from the information given. 

You can, however, tell how many dollars worth of stamps you have. 

50 x 20¢ is the same as 50 x 0.20 $.     Do you see what I did there? 

                                                              $0.20 is just another way to write 20¢. 

50 x $0.20 = $10.00     You have ten dollars worth of stamps. 

_.

Two consecutive integers a and b, where a is bigger, have a sum of 7.  Find a^2 - b^2.   

I must be missing something here, because it's obvious that

4 and 3 satisfy the requirement to be consecutive and total 7. 

So 4² – 3²   =  16 – 9   =   7                    What am I missing?

_.

Guest  had everything correct  except the last sentence.....the height  of the first pole =   5 meters instead  of 20 meters

This one  is tricky  because  there  are  no "numbers"....but....no problem  !!!!

a^2  - ap + p^2  - 3q             assuming  "a"  is the variable  and p, q  are  constants

Write

a^2  - pa  + p^2  - 3q         

Take (1/2)p  ..... square  it   =  p^2  / 4    .....add  and subtract  it  

 ( a^2  - pa  + p^2/4 )   +  p^2  - 3q - p^2/4    

Factor the  1st three terms....simplify the  rest

(a - p/2)^2  + (3/4)p^2 - 3q

The vertex is   (p/2 , (3/4)p^2 - 3q )

Let  H  be the height of the bldg  and D  be  the original distance from the  buildng

So

tan 60  =    H  / D 

cot 60 =  D /H

H cot 60  =  D

And we have that

tan   45  =   H  / ( D + 40)

  1  =  H / ( D + 40)

D + 40  =  H

H cot 60 + 40  = H

40  = H ( 1 - cot 60)   

40 / (1 - cot 60) = H  ≈  94.64  ft

Let  Bob's  speed  =  B

Then Frank's speed    = 2B

So     Rate * Time  =  Total Distance.....so.......

B*4   +  (2B)*(4)   =  480    simplify

4B +  8B  = 480

12B  = 480    divide both sides  by 12

B  =  40 km /hr  = Bob's speed

2B  = 80 km/hr  = Frank's speed

3.

From the results  in   (2)   we  would expect  that

p(1 + 2i)  =  1 - 3i       and

p(3 - 4i)  =  2  + i

p ( 2 + 4i)    =    ?      (cannot  be  determined  )

2.

p(2 + i)   = (2 + i)^3  + 2(2 + i)^2  - (2 + i)  + 4

              =  (2 + i )  [  (2 + i)^2  + 2 (2 + i) - 1 ] + 4

              =  (2 + i)  [ 4 + 4i + i^2  + 4  + 2i  - 1  ] + 4

              =  (2 + i) [ 4 + 4i  - 1  + 4  + 2i  - 1  ]+ 4

             =  (2 + i) [ 6 + 6i ] + 4

              =    12 + 6i + 12i  + 6i^2  + 4

              =    12  + 18i  - 6  + 4

               =    10 + 18i

P(2 - i)  =  (2 - i)^3  + 2(2 - i)^2  - (2 - i)  + 4

             =  (2 -i)  [ (2 - i)^2  + 2(2 -i) - 1  ] +  4

             =  (2 -i)  [ 4  - 4i  + i^2  + 4 - 2i - 1  ] + 4

            =  ( 2 -i) [ [ 4  - 4i   - 1  + 4  - 2i  - 1  ]  + 4

             =  ( 2  - i) [ 6 - 6i ] +  4

             =   12 - 6i - 12i  + 6i^2  + 4

              =   12  - 6i  - 12i  - 6 +  4 

              =   10  - 18i

P(1 + 3i)  =  (1 + 3i)^3  + 2(1 + 3i)^2  - (1 + 3i)  + 4

                =  ( 1 + 3i) [ ( 1 + 3i)^2  + 2(1 + 3i) - 1  ] + 4

               = (1 + 3i)  [ 1 + 6i + 9i^2 + 2 + 6i - 1 ] + 4

               =  (1 + 3i) [ 1 + 6i - 9 + 2 +6i - 1  ]+ 4

                 = ( 1 + 3i)  [ 12i -7 ] + 4

                 = ( 12i + 36i^2 - 7 - 21i)  + 4

                 =  ( -9i - 36 - 7 ] + 4

                  = - 39  - 9i

Note  that   P(2 + i)  and P(2 - i)   only differed by signs  on  the  "i"  term

So....we would expect that  P( 1 - 3i)    would  just be    -39 + 9i

 X : 20 where as 8 : 32

8 : 32 = 1 : 4

We make these ratio's into fractions...

1/4 = x/20

We use the proportion laws...

4x = 20

4x/4 = 20/4

x = 5

Thus the vertical pole's length is 20 meters