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I think I get what you're talking about:

Do you actually want to try all of the possible routes? I mean, sure, it makes sense with 4. But even there, you are doing 24 calculations. And this number goes up into the stratosphere pretty darn fast. This is one instance of a mathematical function called a factorial. To take the factorial of a number, you use the exclamation mark, and the function works like so: \(x! = x \times (x - 1) \times (x - 2) \times (x - 3) \times \dots \times 3 \times 2 \times 1\). At numbers like 30 or so, this is impractical for human calculation; there are 6,227,020,800 different routes you could take!

\(n-(-5)=n+5\). When subtracting negative integers, you add them, as \(a-(-b)=a+(-1)(-b)=a+b\).

Doing that, we get \(n+5=35\), \(n=30\). 

Ah yes, I was taking AB to be different from BA. I am not sure if order counts in the initial question or not. ....

Yes that is correct.

0.9 repeater  = 1

You illustrated why using multiples of 0.1 repeater

0.1 repeater =1/9    so    0.9 repeater =9/9 = 1

Here is another illustration.  

0.3 repeater =1/3

0.6 repeater =2/3

0.9 repeater = 3/3 = 1

Note something that we need to be aware of, helperid

If  x  =  -1/2....we will have negatives  under  both radicals....but each side will evaluate to the complex answer, "i"

Try looking up the answers online. Search "(Textbook name) (Chapter number) (Problem) answer"

Thanks, Hectictar.....!!!!

Here it is

The perimeter is given by

P  =  2 ( W + L)

P / 2  =   W + L    ⇒   P/2 - W  =  L     (1)

And the area is

A =  L  * W        sub  (1)  for L

A =  (P/2 - W) * W        simplify

(P/2) W  - W^2     rearrange

-1W^2  + (P/2)W

a =  -1    b  =  (P/2)

Again....the  "W"  that maximizes  the area is given by     -b / [ 2a]  =

- ( P/2) / [2 ( -1) ]  =  - P/ -4  =  P /4

And when  W  =  P/4       L  =  P/2 - P/4  =  P/4

So.....  W = L   =   P/4  =  100 / 4  =   25  ft

This will always be true, AT......for any perimeter......area is maximized  when  W = L  = P/4

Thanks Hectictar!

For \(\sqrt{2x}=\sqrt{4x+1}\):

Square each side to remove the meddlesome square root:

2x = 4x + 1

Isolate x (subtract 4x from each side, since 4 is the coefficient of x on the side with the extra number):

-2x = 1

Divide each side by -2 (the coefficient of x):

\(x = -\frac{1}{2}\)

Let's call the length of the rectangle  L, and the width  W .

perimeter  =  L + L + W + W

perimeter  =  2L  +  2W

100  =  2L  +  2W            Solve this for  L .

100 - 2w  =  2L        Divide through by  2 .

50 - W  =  L

area  =  L * W           Substitute  50 - W  in for  L .

area  =  (50 - W) * W

area  =  50W - W²

area  =  -W² + 50W

The maximum area will be when  W = - 50 / [2(-1)]  =  25

and L  =  50 - W  =  50 - 25  =  25

Google says the formula for momentum is\(p=mv\), where p = the momentum of the object, m is the mass of the object, and v is the velocity of the object. Velocity is just speed with a given direction, so we can solve for speed in place of velocity for here, but not everywhere. The formula for speed is\(s=\frac{d}{t}\), where s is the speed of the object, d is the distance traveled, and t is the time it takes to move that distance. Using the given formula, we can solve for speed:

3.4 meters/2.3 seconds = about 1.47826087 meters/second

Since momentum is measured in kilogram-meters/second, we need to convert 46 grams to kilograms:

46 grams/1000 (conversion rate) = 0.046 kg

Now we combine the 2 outputs:

1.4782608695652174*0.046 = 0.068 kilogram-meters/second.

*Note: If you use the calculator with the decimal, you get the answer, a bunch of 0's, then 4, because the decimal is not exactly equal to the fraction given, however, if the fraction is used, you get exactly 0.068, because the 0.046 cancels out the 2.3 on bottom.

It's fine. You don't have to if you don't feel like it. I'm sure someone else on here can explain it.

I'm a little tired......but the area will be maximized when the rectangle is a square with a side of 25

I'll go through it, AT, if you want me to........

Lateral surface area  =

pi * ( sum of the radii) * sqrt [  difference of radii^2  + height^2 ]  =

pi *  ( 6 + 3) * sqrt [ ( 6 - 3)^2  + 4^2 ]  =

pi * ( 9 ) * sqrt [ 3^2 + 4^2 ]  =

pi * 9 * sqrt [ 25]  =

pi * 9 * 5  = 

45pi   in^2

Ok. Thanks a ton CPhill!

Yes! Thanks so much CPhill!

See your other post for my explanation.....

Sorry.....I should have explained that....!!!!

In a quadratic that has the form  ax^2 + bx + c, the  maximum  [ or minimum]  is produced at the x value  given by :

-b  / [  2a]

So  ......using our function    -1000x^2 + 20000x + 1500000

a =  -1000      and b  =  20000

So....the x value [ additional employees hired ]  that produces the max revenue  is

-20000 / [ 2 ( -10000) ]  =    -20000 / -2000  = 10

Does that help ???

Thanks CPhill, but where did you get the x =  -20000 / [ -2 * 1000]?

Where did you get x =  -20000 / [ -2 * 1000]?

I don't have enough information. Is there anything else that might have been in the question?

I just answered....see your other post, AT