All Answers 352639 Answers

3^(x + 1)* 2  =  108     divide both sides by 2

3^(x + 1) =   54            take the log of both sides

log 3^(x+ 1)   = log 54      and we can write

(x + 1) log 3   = log 54         divide both sides by  log 3

x + 1  =   log54 / log 3       subtract 1 from both sides

x  = log 54 / log 3  - 1  ≈  2.63

However.....if this is supposed to be

3^[ (x + 1) * 2 ]   =  54

3^( 2x + 2)  =  54

3^(2x) * 3^2   = 54

3^(2x) * 9  = 54      divide both sides by  9

3^(2x)  =  6       take the log of both sides

log 3 ^(2x)  = log 6

(2x) log 3   = log 6

2x  = log6 / log 3

x =  log 6 / [2 log 3]

x = log 6 / [ log 3^2]

x = log 6 / log 9  ≈  .8155

Let's call a \(2 \times 2\) square a 'block' and a \(2 \times 1\) rectangle consisting of a column of two \(1 \times 1\) squares a 'blank'. So a \(2\times 10\) grid, for instance, consists of 10 consecutive blanks, numbered 1 to 10 from left to right, and a block occupies two consecutive blanks, say blanks 1 and 2, which I will denote by (1, 2). The max number of blocks that can be arranged on our \(2 \times 10\)grid is, of course 5, and the places they would occupy  would be denoted as (1, 2), (3, 4), (5, 6), (7,8), and (9,10). We will first look at the number of arrangements on the \(2 \times 10\) grid and then extend the results to a \(2 \times 50\) grid.

First note that a block can occupy a \(2 \times k\) blan grid in k - 1 different ways. For instance, on a \(2 \times 10\) blank grid, the block can be placed on (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (7, 8), (8, 9), or (9, 10).

The following table shows that the number of arrangements for a \(2 \times 10\) grid is

\({10 \choose 0 } + {9\choose 1} +{8\choose 2}+{7\choose3}+{6\choose 4}+{5\choose5}=89\)

Thiscan be generalized to a \(2 \times 50 \) Grid. see below

Here’s a copy of the original question.

If x^2+bx+9 has two non-real roots, find all possible values of b. Express your answer in interval notation.

***************

You know you are not supposed to post AoPS questions. You chose to do so, anyway. You then delete your questions because you fear getting caught, or you do not want anyone else to benefit from the solution.  This makes you an APoS (A Piece of Shit) student.

wrnog

Maybe it is wrong but you are expected to say why you believe it to be so.

Maybe you are just a stupid brat who has recently learned how to spell 'wrong' and you are trying to impress the world with your great knowledge.

Considering that you have written 'wrong' as a response to a couple of answers very recently I expect my explanation of your motives sounds very likely.

If you want to say  an answer is wrong guest you are expected to say why you believe this to be so.

That seems a reasonable starting point dragon.    

I do not know if you did it yourself but it is a better start than i had when I tried to do it with no hints.

I had no idea where to start.

However:

I am not convinced about your continued logic.

Just for one of these configurations I think you have to start by placing one number. I chose to place one of the A which is a multiple of 3

So I think it would be  2! * 3! * 3!   = 24 ways

But you have

C B A C B A C B A          YES THAT WILL WORK   24 ways

B C A B C A B C A          THAT WILL WORK TOO    24 ways

I can't see any other way to do it.  But maybe you can.  

So I only get 48 ways but I could easily be wrong.

wrong

Why did you repeat your question? Your example at the end is not correct –it’s absurd.

The total should shows four or five places beyond the decimal and you'll have to round up or down. You will explain who keeps the difference, the owner or employee after you round it back to 2-places beyond the decimal.  This is similar to the price you pay for gas 1/10 of a cent below a break point.  You won't find the tenths of a cent on your paycheck, so where does it go?

The quick answer to your question “where does it go,” “who keeps the difference,” is simple: The fractional payments are distributed in other pay checks. No single person or entity “keeps” the fractional parts of the payments.  To clarify, the payments rounded down are used for payments that are rounded up.

Converting US dollars to Pesos

The exchange rate is $0.0529 per Peso. So it’s necessary to divide the dollars by the exchange rate to convert to Pesos.

For your example: (40) * ($19.55/hour) = $782.00

($782.00) / (0.0529) = 14782.608 Pesos

In finance (payables), the value of the third decimal is used to determine if the second decimal is rounded up or remains the same. If the third decimal is five (5) or higher, then the second decimal is incremented. If third decimal is four (4) or lower, then the second decimal remains as it is. Using this criterion, the pay check for this employee is 14782.61 Pesos; it’s rounded up to the next centavo.

Theoretically, half the time this third decimal will range from (5) to (9) and the half the time it will range from (0) to (4).  Thought this is true, when the third decimal is zero (0), there is no fractional value, so it does not generate any excess. The fraction that is collected for 4, 3, 2, and 1, pays for 9, 8, 7, and 6; but the zero (0) does not pay for the five (5).  This sounds trivial, but after a million paychecks, the company pays an extra 1000 Pesos –about $52.92.   

This confirms the old maxim: Watch your centavos closely, the pesos will take care of themselves.  

GA

The triangle ABC is made up of three smaller triangle, namely AIC, AIB, and BIC. These triangles have the same height, namely r, the inradius of the triangle, and their bases are \(a, b\ and\ c.\) Adding the areas of these triangle we get the area of the triangle ABC, and in terms of r, that area is

\(\frac{1}{2}ar+\frac{1}{2}br+\frac{1}{2}cr=\frac{a+b+c}{2} \cdot r\) \(=\frac{6+8+10}{2} \cdot r=12r\).

So if we can find the area of triangle ABC , then by dividing that area by 12, we would have the inradius. Fortunately Heron, or Hero of Alexandria, some Greek dude who lived around 10 AD, left us a formula by which we can calculate thee area we need using only the known sides of the triangle ABC. If we let \(s=\frac{a+b+c}{2}\), a quantity we calculated to be 12, the the area of ABC would be\(\sqrt {s(s-a)(s-b)(s-c)}\)\(=\sqrt{12(12-10)(12-8)(12-6)} = \sqrt{12\cdot48} =\sqrt{36\cdot16}=24\). So the measure of the inradius is 24 divided by 12 or 2 units of length.

It would depend on how the sting is getting streched. 

As guest commented, if it is getting streched sideways it would make no difference.  If it was getting streched from the back and the front stayed put then the ant is going backwards i think.

If from both ends then I think the ant would probably keep going backways.  

Only if it is getting streched in a forward direction will it add to the ants progress.

It would have been good if you had added in the address of your teachers desmos display.

at least next year they give us 75 min.

but problem 8 is like problem 25 on the amc 8

ok BM I like that.

That is worth thinking about :)

Htyhang919     You are one extremely rude person!

You delete your question after Hectictar has put in all that effort to answer it .   How incrdeibly RUDE.

I see you have done it on your other questions too.

I guess you have been caught cheating or you are frightened that you will be. 

JUST HOW RUDE !!!

This thread is comedy GOLD! 😂 Anyways, I'll tell you the thing you were missing. When the string stretches, the point the ant was on moves with it. So in theory, the ant will travel a little more than an inch and will eventually overtake the string! You can even view this in Desmos for all you visual learners.......at least that's what my teacher did! 😂 But most people do actually think that way; the ratio will always stay constant; but you must think outside the box for this one!

Good job everyone!

BasicMaths

The first step in this is to separate \(a^2\) from the rest of the function.

\(a^2-3a+9=0\) becomes \(a^2=3a-9\).

Now there is just one step to find \(a^3\). Can you take it from here?

AoPS. Morrivillle,

Try not to post homework questions.

I think everyone would have-

T-T

I DIDN'T EVEN GET TO 25! OR 20!

MY NERVES FAILED ME AND I HAD A NOSEBLEED IN THE MIDDLE AND I RAN TO THE NURSE AND RAN BACK AND THAT TOOK 10 MIN AWAY

What did happen?

Don't worry about it, the same here!

I would think that if I had more time, I would've done better.