CPhill

7.  The numbers 19, a, b, c, 73 form an arithmetic sequence, in that order.
Find a+b+c.

19 + 4d  =  73       subtract  19  from both sides

4d  =  54   divide both sides by 4

d  =  13.5

Calling m the first term  ....the sum  of a + b + c  is

( m + d ) + (m + 2d) + (m + 3d)  =

3m  + 6d  =

3(19) + 6(13.5)  =

57    + 81  =

138

Sorry, ant....I would just be guessing on the rest and that usually leads to wrong answers....maybe some other people know the solutions to the rest of these   !!!!

1. Paula can paint a room in six hours. Paula starts painting the room at
9:00 AM. At 11:00 AM, Carlos accompanies Paula, and both of them,
working together, finish painting the room at 1:30 PM. Assuming both
painters always paint at a constant rate, how many hours can Carlos
paint the same room working by himself?

Paula paints 1/6 of the room every hour

By the time Carlos joins in, Paula has painted   2 (1/6) =  1/3 of the room

And after Carlos joins....she paints an additional 2.5 hrs

So...by  1:30 PM....she has painted  1/3  + (2.5)(1/6)  =  1/3 + (5/2)(1/6)  = 

1/3 + 5/12  =  4/12 + 5/12  = 9/12  =  3/4  of the room

So.....Carlos  must paint  the other  1/4 of the room in  2.5 hours

So...the time it would take him to paint the room by himself must be 4 times this  =

10 hrs

LOL!!!.....yep, that helps  !!!

x^4 + 7x^3  + 9x^2  - 17x - 20

The possible rational zeroes are  ±  ( 1, 2, 4, 5, 10 and 20)

We have  one sign change ..so we have 1  positive root 

To  find the number of possible negative roots look at f(-x)  =

(-x)^4 + 7(-x)^3 + 9(-x)^2 - 17(-x) -20  =

x^4 - 7x^3 + 9x^2 + 17x - 20

We have hree sign changes so we either have 3 or 1  negative roots

Let's write the above in a slightly different manner

x^4 + 7x^3 + 6x^2 + 3x^2 - 17x - 20        factor this

x^2(x^2 + 7x + 6)  +  (3x - 20)) (x + 1)  =

x^2 ( x + 1) (x + 6) + (x + 1) (3x - 20)   factor out the ( x + 1)

(x + 1)  [ x^2 (x + 6) + (3x - 20)  ]

(x + 1) [ x^3 + 6x^2 + 3x - 20 ]  the second polynomial factors as  (x + 4) (x^2 + 2x - 5)

So   we have

(x + 1) (x + 4)(x^2 + 2x - 5)

Setting the first two  factors to 0  and solving for x produces   x = - 1   and x =  4

The other two roots  come from using the quadratic formula to solve the 3rd polynomial...these are

x  = √6 - 1    (p0sitive)   and  x  =    -√ 6  -  1   (negative)

So....we have 1 positive  root and 3 negative roots

For which polynomials ??/

8)ABCD is a cyclic quadrilateral. Angle A, angle B, and angle C form an arithmetic sequence in this order. What is angle D in degrees?

What is the order of the arithmetic sequence   ???

9)Points A and B are on circle O such that arc AB is 80 degrees. A circle is constructed that passes through A, B, and O. Find the measure of arc AOB on this circle.

AOB  in the smaller circle is also 80°

So.....it's intercepted ard  is  twice this  = 160°

1)Given angle APS = 77 degrees and arc AP = 123 degrees, find the degree measure of arc PS.

If angle APS  = 77°, then  arc AS is twice this = 154°

So...arc PS  = (360 - 123 - 154)   =   83°   

2)In the figure below, EF is a diameter of the circle. Arc AE = 35 degrees and arc CF = 80 degrees. What is the measure of angle ABC, in degrees?

Angle AFE  =  (1/2)(35°)  = 17.5°

Angle CEF  = (1/2)(80°)  =  40°

So ....angle EBF  = angle ABC  =  (180 - 17.5 - 40)   = 122.5° 

3)In the diagram, angle A = 30 degrees, arc DE = 170 degrees, and arc BC = 110 degrees. Find the measure of arc CE, in degrees.

The sum of arcs CE + DB  = (360 - 110 - 170)  = 80”

And  angle A  = (1/2) (arc CE - arc DB).....so....

30  = (1/2)(arc CE - arc DB)

60 = CE - DB

So

CE + DB  =  80

CE - DB   =  60      add these

2CE  =  140     divide both sides by 2

arc CE  = 70°

4)In circle O, AP is a diameter of length 17 and PS is chord of length 8. What is the length of chord AS?

AS  will form a leg of a right triangle with AP  the hypotenuse and PS the other leg....so

AS  =  √ [ 17^2  - 8^2 ] =  √ [ 289 - 64 ]   =  √ [ 225]  =   15 units

5)Grogg draws the following figure. It so happens that angle SAP is 174 degrees less than 4 times angle SOP. Find the degree measure of angle SAP.

SAP + 174  = 4*SOP

But SAP  and SOP  intercept the same arc PS...so they are equal

So...substituting

SAP  + 174  = 4*SAP   subtract SAP from both sides

174   = 3*SAP         dvide both sides by 3

58° = SAP

6)In the diagram, minor arc AB : minor arc BC : major arc CA = 1 : 3 : 5. What is angle ABC in degrees?

There are  1 + 3 + 5  =  9 equal arcs in the circle....and angle ABC  spans 5/9  of these

So  major arc  CA  =  (5/9) (360)  =  (360/9) * 5  =  40 * 5  =  200°

And since angle ABC is an inscribed angle intercepting this arc, it is 1/2 of this  = 100°

7)In the figure, if MR = MK, the measure of arc MK is 130 degrees, and measure of arc MQ is 28 degrees, then what is angle RPK, in degrees?

Since MR = MK, then arc MR  = arc MK,,,,and the sum of these arcs is 260”

So minor arc RK   =  360 - 260  = 100°

So...since angle RMK intercepts this arc it has 1/2 of its measure  = 50°

And since arc MQ is 28, then angle QKM  = 14

So in triangle PKM angle KPM  =  180 - 14 - 50  = 116°.....and angle RPK is supplemental to this  = 180 - 116  =  64°

Thanks, guest...14 is correct....

1. Four friends agree to split the cost of a rental car equally. One of the friends decided to take a taxi, so the remaining three friends each paid $11 more. What was the cost of the rental car, in dollars?

Let the cost of the rental car = C

So....the amount that each would pay before one left is  C / 4

So.....after one lleft.....the amount each pay is  C/4  + 11

So

3 (C/4 + 11)  =  C       simplify

(3/4)C + 33  = C

33  =  1/4C     multiply both sides by 4

$132   =  C 

Notice that  the 23rd person can shake hands with 22 other people

And the 22nd person can shake hande with 21 people [ this person has already shaken hands with the 23rd person]

And the 21st person can shake hands with 20 people[ this person has already shaken hands with the first two ]

So....the total possible handshakes are

22  +  21  + 20  + ......+  1  =

 1 + 2 + 3 +  ......+ 20 + 21 + 22

Which is just the sum of the first 22 positive integers and is given by :

(22)(23) / 2   =   11 * 23  =  253 handshakes