All Answers 356330 Answers

[EBA] = 1/6[ABC]

Let the polynomial $P$ be defined by $P(x)=x^3+bx^2+cx+d$.   Now substitute $\sqrt[3]{4}+1$ for $x$ in the formula for $P(x)$ to get $P(\sqrt[3]{4}+1)$ and collect terms so that you have
\[
    P(\sqrt[3]{4}+1) = k\cdot 4^{1/3} + \ell\cdot 4^{2/3} + m
\]
where $k, \ell, m$ are integral expressions involving $b,c,d$.  Now set up three equations $k=0$, $\ell=0$, and $m=0$, and solve for $b,c,d$.   You should get the solution $P(x) = x^3-3x^2+3x-5$.   (You should check that $P(\sqrt[3]{4}+1) = 0$.)

In polar coordinates:  \(z=re^{i\theta} \text{ and } \bar{z} = re^{-i\theta}\)   so \((\bar{z})^n=(re^{-i\theta})^n=r^ne^{-in\theta}\) and \(\bar{z^n}=\bar{(r^ne^{in\theta})}=r^ne^{-in\theta}\)

The smallest possible value of n is a four-digit positive integer

\(\begin{array}{|rcll|} \hline \mathbf{n} &=& \mathbf{59 + 315k} \\ 59 + 315k &>& 999 \\ 315k &>& 999 - 59 \\ 315k &>& 940 \\\\ k &>& \dfrac{940}{315} \\\\ k &>& 2.9841269841 \\ \mathbf{k} &=& \mathbf{3} \\ n_{\text{The smallest possible value of n is a four-digit positive integer }} &=& 59 + 315*3 \\ \mathbf{n} &=& \mathbf{1004} \\ \hline \end{array}\)

81 is only a little number.

There are only 8 perfect squares that are smaller.

Surely you can work this out with trial and error!

0.01010101 = 1/99

0.02 = 2/100

1/99 + 2/100           (100 + 198) / 9900 = 149 / 4950

The number a in such context is always taken to be a nonnegative integer less than m.  If you find a solution that is negative, do add an appropriate multiple of m to it to make it nonnegative and less than m.  E.g. if you find a solution of -2 for a, and your m is 5, then add 5 to a to get -2+5 = 3.  So your a is actually 3.   So a+m in this case is 3+5 = 8.

cosC = 4/7

Hi catmg,

It is your original assumption that I am having trouble with.

"I'm going to assume that n only has factors of 5 and 11. "

If there was no answer before yours, why would you assume this.

You do not need to answer.  (unless you really want to) I am just airing my thoughts.

Isn't N a four-digit positive integer?

Your pic is not displaying.

Human moderators have not blocked these images, nor can we see them or unblock them.

They have been blocked automatically.  Maybe they are copyrighted?

Convert into fraction:  \(0.1 \overline{83}\)

Hello Guest!

\(1000\cdot 0.1\overline{83}=183.\overline{83}\)

 \(-\underline{10\cdot0.1 \overline{83}\ =-1. \overline{83}}\)

   \(990\cdot 0.1\overline{83}=182.\overline{00}\)

\(0.1\overline{83}=\dfrac{182}{990}=\color{blue}\dfrac{91}{495}\)

  !

I do thank you.  I will still have to think about it but I think your newest explanation will help a lot.

And I am glad that you are enjoying yourself.   

It is great to have people, like yourself, properly interacting here.

For others: 

You do not need advanced maths to interact properly here. 

A number of younger people, or less mathematically advanced people, are greatly valued members of this forum   

Thanks very much Bginner.

You have given me homework.

Fair enough, I will take a look a little later :)

oh, you should give yourself a point when you answer questions.  That is what I think anyway :)

Formula Solution for Finding Arc Length: 2*pi*r(0/360) 

Use the given dimensions to find the length of each circular arc. (Round your answers to two decimal places.)

a)   a= 60, C= 18cm                                             

     = 2*pi*r(0/360)     Write the formula for Arc Length.

     18 = 2*pi*r           Substitute 18 for 2*pi*r. 

     = 18(60/360)        Substitute 60 for 0/360. 

     = 3                      Use a Calculator

b) a = 76, r= 4 ft.

    = 2*pi*r(0/360)      Write the formula for Arc Length

    = 2*pi*4(76/360)     Substitute 4 for r and 76 for 0/360.

    = 5.30 ft               Use a calculator.

C) a= 145°, C= 53 in

   = 2*pi*r(0/360)       write the formula for Arc Length

   53 = 2*pi*r           substitute 53 for 2*pi*r.

   = 53(145/360)        Substitute 145 for 0/360.

   = 21.35                 Use a calculator.

d) a= 150°, r= 9m

   = 2*pi*r(0/360)      Write the formula for Arc Length

   = 2*pi*9(150/360)    Substitute 9 for r and 150 for 0/360. 

   = 23.56 m              Use a calculator. 

0.4888... = 22/45

0.3777... = 17/45

22/45 + 17/45 = 39/45

sin(7x) = cos (3x − π/14) + 2

cos(7x - pi/2) = cos(3x − π/14) + 2

Note that cos(a) has to return a number from -1, to 1. 

So the only way this equation is possible is if cos(3x − π/14) = -1 and cos(7x - pi/2) = 1. 

cos(3x − π/14) = -1

cos(7x - pi/2) = 1

Can you take it from here?

=^._.^=

No, this is a calculus question.  

I understand now why you appeared to ignore my initial answer.  You had no idea what I was talking about.    

Well, will it help to know that the desired smallest positive x is in the interval (1.1, 1.2)?

We'll set up an equation by "counting the same thing in two different ways."  Consider the process of tracing out the star by following the five segments in this order a-c, c-e, e-b, b-d, d-a and adding up all the numbers we encountered in the process.

One way of adding the numbers goes like this: Each time we draw a segment we encounter four numbered points.   The picture clearly shows that each numbered point is the intersection of two segments.  So each number is added to the total exactly twice.  So the total sum we get is 2(a+b+c+d+e) + 2(2+3+4+5+6).

Another way of adding the numbers goes like this: We draw the stars by drawing five segments and each segment encountered contains four numbers adding up to 28.   So the total sum we get is 5*28.

The sums we get in two different ways must be the same, so now we can set up an equation

    2*(a+b+c+d+e) + 2*(2+3+4+5+6) = 5*28.

Solving, we get

    a+b+c+d+e = (5*28 - 2*(2+3+4+5+6)) / 2.
 

∠A = 65º

∠B = 70º

∠C = 95º

∠D = 130º

a + b + c = 2 + (-1) + 3 = 5.

We want angle CBD < 45.  Since angle ABC = 60, the probability is 45/60 = 3/4.

1/a^2 + 1/b^2 = 13/7.