All Answers 354377 Answers

ok, 

All of that is really good.  You are really starting to get the hang of it.

You do need to know automatically that if

y=asinx  +b

then 

y intercept of b

midline y=b

Amplitude 2pi

And it will go up first if 'a' is positive and down if 'a' is negative.     (from the x axis)

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Answer the questions for the 2 graphs below.  AFTER you determine the equation, check it using Desmos.

Q1)

a) midline

b) y intercept

c) wavelength

d) amplitude

e)  Does it start going up or down?

f) SO what is the equation

Question 2)

a) midline

b) y intercept

c) wavelength

d) amplitude

e)  Does it start going up or down?

f) SO what is the equation

The wording is ambiguous and not easy to follow.

Provide a diagram and you might get a response.

Equation: y = 3 -4 sin x    

1. "What is the least confusing way to write this?" 

      I'm not exactly sure what you mean by that, I apologize.

Are you asking about how can this equation can be rearranged? If so, I'm not entirely sure. 

2. "What number does the minus sign belong to?" 

      The minus sign belongs to the four I presume.

3. "What will be the main features of this graph?"

       Once again, not entirely sure what you mean here, I assume you mean the midline, y-intercept, etc?

a) Midline: y = 3

b) y-intercept: 3

c) Wavelength: 2π (From after checking the Desmos graph.)

d) Amplitude: 4

4. "What is the y-intercept?"   

      The y-intercept is 3. 

5. "Is the y-intercept on the centerline?"  

      It seems to be. (From after checking the Desmos graph.)

6. "From the y-axis, will the graph start going up or down? How can you tell this from the equation?" 

      I assume it will begin downwards because of the -4 in the equation. Previously, the addition of the minus sign inflects the graph to move downwards initially instead of upwards into the positives. (Accurate after having checked the Desmos graph.)

\(\displaystyle \sin x = 2/3, \text{ so }\cos^{2}x=1-(2/3)^{2}=5/9, \text{ so }\cos x =\sqrt{5}/3.\) 

\(\displaystyle \sec y = 5/3, \text{ so } \cos y = 3/5, \text{ so } \sin y = 4/5.\)

\(\displaystyle \sin(x+y)=\sin x.\cos y+\cos x.\sin y = \frac{2}{3}.\frac{3}{5}+\frac{\sqrt{5}}{3}.\frac{4}{5}=\frac{6+4\sqrt{5}}{15}\approx 0.99628.\)

The probability is 22/27.

The probability is 37/44.

The probability is 14/81.

Neat question.

(a+b) n can be written as

Sum[k={0,n} Choose(n,k) a kb n-k]

i.e. Choose(n,0) b n+Choose(n,1) a b n-1 + ... Choose(n,n-1) a n-1b + Choose(n,n) a n

Now you're probably thinking ok wth is "Choose"?

This is the binomial coefficient, "n choose k" and they are the values of a given row in Pascal's triangle. Come down n rows and go over k elements in the triangle and you get the Binomial coefficient corresponding to "n choose k" = Choose(n,k) = Binomial(n,k) (different names for the same thing).

So in this case if you look at the n=5 row (really the 6th row, the first row is n=0) of Pascal's triangle you see it's

1 5 10 10 5 1

in other words

Choose(5,0)=Choose(5,5) = 1
Choose(5,1)=Choose(5,4) = 5
Choose(5,2)=Choose(5,3) = 10

so (a+b) 5 = 1*a 5 + 5 a 4 b + 10 a 3 b 2 + 10 a 2 b 3 + 5 a b 4 + 1 b 5

now let a=3t and b=4 and substitute those into the formula above and simplify things and you're done

||2a - 3b|| = 2*sqrt(13).

O is the centre and R the radius of the circle circumscribing the triangle ABC, AO, BO, and CO meet the opposite sides in L, M, N respectively.

Show that

\((i)\quad AL = \dfrac{ b\sin C} { \cos(B-C) }\)

\(\text{Let $\angle BAO= \angle ABO = w$ } \\ \text{Let $\angle CAO= \angle ACO = v$ } \\ \text{Let $\angle CBO= \angle BCO = u$ } \\ \text{Let $\angle A= v+w$ } \\ \text{Let $\angle B= w+u$ } \\ \text{Let $\angle C= u+v$ } \\ \text{Let $\angle CLA = \varphi$} \)

\(\begin{array}{|rcll|} \hline A+B-C &=& (v+w)+(w+u)-(u+v) \\ A+B-C &=& 2w \quad | \quad A+B = 180^\circ - C \\ 180^\circ - C-C &=& 2w \\ 180^\circ -2C &=& 2w \quad | \quad : 2 \\ 90^\circ -C &=& w \\ \mathbf{w} &=& \mathbf{90^\circ -C} \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline A-B+C &=& (v+w)-(w+u)+(u+v) \\ A-B+C &=& 2v \quad | \quad A+C = 180^\circ - B \\ 180^\circ - B-B &=& 2v \\ 180^\circ -2B &=& 2v \quad | \quad : 2 \\ 90^\circ -B &=& v \\ \mathbf{v} &=& \mathbf{90^\circ -B} \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline -A+B+C &=& -(v+w)+(w+u)+(u+v) \\ -A+B+C &=& 2u \quad | \quad B+C = 180^\circ - A \\ 180^\circ - A-A &=& 2u \\ 180^\circ -2A &=& 2u \quad | \quad : 2 \\ 90^\circ -A &=& u \\ \mathbf{u} &=& \mathbf{90^\circ -A} \\ \hline \end{array}\)

In triangle ALC:

\(\begin{array}{|rcll|} \hline \varphi &=& 180^\circ-(v+C) \quad | \quad \mathbf{v=90^\circ -B} \\ \varphi &=& 180^\circ-(90^\circ -B+C)\\ \varphi &=& 180^\circ-90^\circ -(-B+C)\\ \varphi &=& 180^\circ-90^\circ -(C-B)\\ \varphi &=& 90^\circ -(C-B)\\ \mathbf{\varphi} &=& \mathbf{90^\circ +(B-C)} \\ \hline \end{array}\)

In triangle ALC - sin rule:

\(\begin{array}{|rcll|} \hline \dfrac{ \sin{C} } {AL} &=& \dfrac{\sin(\varphi)}{b} \\\\ \dfrac{ \sin{C} } {AL} &=& \dfrac{\sin\Big(90^\circ +(B-C)\Big)}{b} \\\\ \dfrac{ \sin{C} } {AL} &=& \dfrac{\cos\Big(B-C\Big)}{b} \quad | \quad \updownarrow \\\\ \dfrac{AL}{ \sin{C} } &=& \dfrac{b} {\cos\Big(B-C\Big)}\\\\ \mathbf{AL} &=& \mathbf{\dfrac{b\sin{C}} {\cos\Big(B-C\Big)}} \\ \hline \end{array}\)

Thanks guest and thanks to Max.

Max's answer required very good visual insight. 

I always knew he was clever from when he first started posting.

Unfortunately I am not likely to remember this.  How sad.

Question 1

Jenny, the circumference of a circle is 2pi*r

r stands for radius but it can also stand for radians.

So in a complete circle there are 2pi radians and 360 degrees.

So long as you know that you can work all the others out for yourself.

Thanks Melody.....I think I got it now...hope you and Tiggsy approve. Great thanks!!

Dear all thanks for your help.....solved part 1 as follows:

Hi Melody: See this elegant answer # 11 by Max Wong:  https://web2.0calc.com/questions/help-plzzzz_3

\(\frac{2\sqrt[3]9}{1 + \sqrt[3]3 + \sqrt[3]9}\\~\\ =\frac{2*9^{\frac{1}{3}}}{1 +3^{\frac{1}{3}} +9^{\frac{1}{3}}}\\~\\ =\frac{2*3^{\frac{2}{3}}}{1 +3^{\frac{1}{3}} +3^{\frac{2}{3}}}\\~\\ let\;\; x=3^{1/3}\\~\\ =\frac{2x^2}{1 +x +x^2}\\~\\ =\frac{2x^2}{x^2+x+1}\\~\\ \)

I really do not see this simplifying very nicely

coding:

\frac{2\sqrt[3]9}{1 + \sqrt[3]3 + \sqrt[3]9}\\~\\ 
=\frac{2*9^{\frac{1}{3}}}{1 +3^{\frac{1}{3}} +9^{\frac{1}{3}}}\\~\\
 =\frac{2*3^{\frac{2}{3}}}{1 +3^{\frac{1}{3}} +3^{\frac{2}{3}}}\\~\\ 
let\;\; x=3^{1/3}\\~\\ 
=\frac{2x^2}{1 +x +x^2}\\~\\ 
=\frac{2x^2}{x^2+x+1}\\~\\ 
 

Hi OldTimer,

Ok so I was looking at it correctly after all.....

Hi Melody.....Apologies....you are right of course. I got distracted by the Windows graphics and loading process to site. Will amend for correctness sake.  

Thanks Tiggsy.     

Mine was completely correct.

I did not give you the final answer.   I only gave you P(pop)  

You finished it yourself, which is great,  but you should have understood that my answer was purposefully not complete.

Billy and Bobbi each selected a positive integer less than 200. Billy's number is a multiple of 18, and Bobbi's number is a multiple of 24. What is the probability that they selected the same number? Express your answer as a common fraction.

200/18 = 11.1111111111111111                  integer part 11

200/24=  8.3333333333333333               integer part 8

factor(18) = 2*3^2

factor(24) = 2^3*3 

lowest common muultiple = 2^3* 3^2 = 8*9 = 72

So there is 2 commmone factors less than 200,       72 and 144

Probability that they both chose 72 = 1/11 * 1/8 = 1/88

Probability that they both chose 144 = 1/88

Probability that they both chose the same number = 2/88   =  1/44

Answering guest:  Why do you think that this is incorrect?

I just read this from Google

"Tracheids are present in all vascular plants whereas vessels are confined to angiosperms. Tracheids are thin whereas vessel elements are wide. Tracheids have a much higher surface-to-volume ratio as compared to vessel elements. Vessels are broader than tracheids with which they are associated.M....

NOW.

If you do not know what an angiosperm is then google it.  

So without checking till afterwards.

y= 3-4sinx 

Q  What is the least confusing way to write this?  What number does the minus sign belong to?

What will be the main features of this graph?

Q What is the y intercept?    Is the y intercept on the centre line?  

From the y axis will the graph start going up or down?  How can you tell this from the equation?  (in the positive x direction of course)

Ok so you know that 16% of 120 is 19.2, so now I'm trying to see what 19.2 is 30% of.

So I have the equation x*0.3 = 19.2, so x = 19.2/.3, which is then equal to 64.

120 - 64 = 56.

The answer is 56.