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Oscar....we don't have any "x" in the function to put "5" into so we can evaluate  f(5)

May be several ways to do this

Notice that the slope of a segment drawn between O and A  =  [ -10- 0] /[10-0]  = -10 /10 = -1

Which is the same slope of the line y = -x + 6

We can find the midpoint of OA  and write an equation of the line through this midpoint that is perpendicular to y = - x + 6

The intersection of these two lines will give us the "P" we are looking for

Midpoint of OA  =  (5, - 5)

Equation of a line through this point perpendicular to y = -x  + 6  is

y = (x -5) - 5

y = x - 10

Find the x coordinate of the intersection of these lines

-x + 6  = x - 10

2x = 16

x = 8

And y = -8 + 6  =  -2

So P  =  (8, -2)

And we can see that this point makes PO = PA

Note that the possible outcomes are

(1,1)    (2,3)   (3,3)  (4,4)

(1,1)    (3,2)   (3,3)  (4,4)

(1,2)    (2,4)   (3,4) 

(2,1)    (4,2)   (4,3)

(1,3)    (2,2)

(3,1)    (2,2)

(1,4)

(4,1)

Number of 1's           Probability

0                                  12/20 =  .60

1                                    6/20  = .30

2                                    2/20 =  .10

1) The product of n counting numbers is 2010. The sum of these counting number is also 2010. What is the greatest possible value of n?

Your question doesn't seem to make sense:

2010 = 2 * 3 * 5 * 67

The SUM of the above 4 prime factors CANNOT = 2010.

However, 2010 has 16 DIVISORS =1 | 2 | 3 | 5 | 6 | 10 | 15 | 30 | 67 | 134 | 201 | 335 | 402 | 670 | 1005 | 2010 .

You can add up 3 of them to make 2010 =1005 + 670 + 335 =2010.

Let the distance that Bob walks east in 10 minutes  =  x

So.....he is x meters from the intersection in 10 minutes

And Alex is  x meters from the intersection after 10 minutes

So....the distance that Alex walks in 10 minutes must be (1200 - x)

So.....in 100 minutes, Bob must be 10x meters east of the intersection 

And  Alex must be the same distance north of the intersection

So....in 100 minutes Alex  walks  10(1200-x) meters......and this distance must equal the 1200 meters to the intersection plus the 10x meters north of the intersection

So

10 (1200 - x) = 10x + 1200   simplify

12000 - 10x = 10x + 1200

10800 = 20x     divide by 20

540 = x

So...the distance that Alex is from the intersection after 100 minutes is

Total distance walked - 1200 m  

10 (1200 - x) - 1200 =

10(1200 -540) - 1200 =  10(660) - 1200 = 6600 - 1200 =    5400 m

Note that Bob  is the same distance from the intersection = 10x =  10(540) = 5400 m

OH GOLLY

According to my count, there are 24 combinations that will give you either a Prime number or a Multiple of 4:

So, 24 / 6^2 =24 / 36 = 2/3 - probability.

First set them equil to each other

2x-2=x^2+2x-3

Subtract 2x from each side

-2=x^2-3

Add 3

1=x^2

Take the square root

x=+-1

Now plug both values of x into the equation to find y

x=1

y=2x-2

Substitute 1 for x

y=2(1)-2

y=2-2

y=0

One of the solitutions is the point (1,0)

x=-1

y=2x-2

Substitute -1 for x

y=2(-1)-2

y=-2-2

y=-4

The other solitution is (-1,-4)

\(\frac{2}{\sqrt[3]{4}+\sqrt[3]{32}} \)     \(\frac{\sqrt[3]{A}}{B}\)

Multiply the top/bottom  by   ( ³√[4^2] - ³√[4*32] + ³√[32]^2 )

2  ( ³√[4^2] - ³√[4*32] + ³√[32]^2 )

_____________________________________

(³√4 + ³√32) ( ³√[4^2] - ³√[4*32] + ³√[32]^2 )

This resolves the demoninator into  4 + 32 =  36

So we have

2 ( 2³√2    -  4³√2 + 8³√2 )                    6³√2           ³√2

_______________________  =         ______  =   _____

                 36                                         18                3

A = 2     B = 3

And their sum is 5

Thanks for your help everyone!!

2. Find the largest value of n such that 3x^2+nx+72 can be factored as the product of two linear factors with integer coefficients.

We need the  max sum of two factors that multiply to 72   and the same for two factors that multiply to 3

These are   1,72     and  1, 3

So......n will be maximized when we have

(3x + 1) (1x + 72)  =  3x^2 + 1x + 216x + 72  = 3x^2 + 217x + 72

n = 217

I don't get this question....

L *  W   =  Area

L *  4  =  (8 + 16x)           divide both sides by 4

L  =  ( 8 + 16x)             8         16x                 

        ________   =      __  +   ____     =     2 + 4x

             4                     4           4

So....the dimensions should be    4 *  (2 + 4x)

Wow thanks Cphill (; 

hello hhh

hello,I hope you can understand me！

WOO-HOO, Nick.....CORRECT!!!

THX, guest....

It's EXACTLY the same problem as Sam Lloyd's  !!!!!

[ I noticed that, too ..... ]

BTW....here's the way to determine the Fermat point :

1) Construct equlateral triangles on two sides of the original triangle

2) Connect the new vertices to the opposite vertices of the original triangle

3) The intersection of these connected segments is the Fermat point

The image shows that the Fermat point in this problem is ≈ ( 4.63, 1.75)

1. The radius of the circle is the discance between P and Q. Plug these into the distance formula:

\(radius=\sqrt{(-3-9)^2+(4--3)^2}\)

\(radius=\sqrt{(-12)^2+(7)^2}\)

\(radius=\sqrt{193}\)

The formula for the area of a circle is \(\pi r^2\). Plug the radius into this to find the area.

\(\pi(\sqrt{193})^2\)

Area=\(\boxed{193\pi}\)

Another way:
Cphill: Does this question remind you of a famous puzzle by Sam Lloyd about a river's width?
Let the total distance between A and B =D
Alex drove 32 km                Bob drove D - 32
When they met the 2nd time they had already covered 3 times the distance between the 2 towns. Let us take Bob's first distance, or D - 32.
3 x (D - 32) - 64(the distance covered when they met the 2nd time) =D - the distance between the 2 towns.Or:
3D -96 -64 =D
2D =160
D = 80 km - the distance between A and B.

Well the triangle is 

4*8=32 

32/2= 16 

and the rectangle is 

13*8= 104   

16+104= 120

so 120 is the answer I belive this is right! 

We have a rectangle  with a width of 13  and a height of 8

And a triangle with  a base of (17 - 13) = 4   and a height of 8

Can you figure it out from this, Nick ???

Thanks 

4 (4x + 2y  + x + y)

Simplify inside the parentheses

4(5x + 3y)     is one form

Distribute  the 4 over the terms in the parentheses  =   20x + 12y   is the other form

So......first answer 

Thanks Cphill