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Distance between  (1, 5)  and  5, 3)  =

sqrt  [  ( 5 - 1)^2 + ( 5 - 3)^2 ]  =

sqrt [ 4^2 + 2^2]  =     sqrt [ 20]  

And if we connect these two points with a chord as well as drawing two radii to each point from the center of the circle, we will have an equilateral triangle....and since this triangle will have equal sides, the radii will have the same length as the chord

So....the area of the circle  =   pi * radius^2    =  pi * ( sqrt(20) ) ^2  =    20 pi units^2   ≈ 62.83 units^2

Angle AOB   = Angle MOC

And angle CMB  = angle ABM

So...by AA congruency....

Triangle BOA  is similar to triangle  MOC

And since  AB  =  AD = 4   and MC  = 2 .....then  the scale factor of triangle MOC to triangle BOA is 2 : 4  =    1 : 2

So

OC : OA   =   1 : 2   =   1 / 2 

Oops!!!....just a typo.....you are correct!!!

I have fixed it....

1.5293

The amplitude will be 1/2 of the variance in day length    2 hrs 53 min x 1/2 = 1 hr 26 1/2 min

The period is 365 days  (in a year)

Here is a graph (and the equation) with March 21st being the origin , x axis is days past March 21st and the y-axis is day length.

1.4416 is 1 hr 26 1/2 min in decimal    .9863 is the degree value of each day(360 degrees  in 365 days)  

   and 12.133   is the decimal equivalent of baseline equinox value of  12 hrs 8 miin

Note:   I am not sure what 'vary by 2 hr 53 min from equinox' exactly means.....if it means the shortest day is 2 53 shorter and the longer day is 2hr 53 longer (which I now think it might) then the amplitude is 2 hr 53 = 2.8833 hr    and  the equation becomes

2.8833 (sin .9863x) + 12.133    and is shown here:

CPhill I think you have a mistake,all it's right but in the end it's 

angle ZCA   =   angle HCA  =   47° NOT   43°

because : 

angle CZA + angle ZAC + angle ZCA = 180° in triangle ZAC

so 43° + 90° + angle ZCA =  180° 

so angle ZCA = 180°-133° = 47° 

So angle ZCA   =   angle HCA  =   47°

Angle BAC = angle ZAC  =  43°

Angle CZA = angle CZB = 90°

So triangle ZAC  is a right triangle and angle  ZCA  is complementary to angle ZAC  = 47°

And angle ZCA   =   angle HCA  =   47°

EDIT TO CORRECT AN ERROR>>>>!!!!

I am not 100% sure, but based on my math the answer would be 1.

I remember this from an earlier question but forget how I did it, and now I can't find out!

Sorry, but again I'm pretty sure it's 1.

You are very welcome

:P

Okay. The phone is free, so that does not cost any money.

The company charges you £10 each month and 5p for every text.

1) Let's see. We know it says "In May," which is just one month. Therefore, you will pay £10 for the month.

However, we are not done! You also send 11 texts and each text is 5p, so to find the cost we do \(5p\times11=55p\).

\(55+10=65\) so the total cost would be 65 pounds.

The answer would be £65.

2) We do the same strategy. Again, we only need to pay for one month as it says "In June." Therefore, you will pay £10 for the month.

Again, we are not done! This time, you text a whopping 80 texts, but we still simply do \(5p\times80=160p\).

\(10+160=170\)so the total cost would be 170 pounds.

The answer would be £170.

3) This is really simple since you don't send any texts, you get only the monthly bill, £10.

The answer would be £10.

4) We always need to pay £10 per month, so we know the "C" part of the equation is actually \(10C\) .

For the texts, you always pay £5 per text, so the text part of the equation is simply \(5T\).

Therefore the equation would be \(10C+5T\)

You are very welcome! 

:P

Answer to 4_39:

It does become fairly complicated with the roll of 10 dice!. However, there is this "general formula" that gives the correct answer to any number of dice => 6. The reason for subtracting and adding is because of "overcounting" and "undercounting" as is discussed extensively here:
https://web2.0calc.com/questions/probability_882
So, the actual formula looks like this:
1 - (6 nCr 1 *(5/6)^n - (6 nCr 2*(4/6)^n) + (6 nCr 3*(3/6)^n) - (6 nCr 4*(2/6)^n) + (6 nCr 5*(1/6)^n)), where n=10 in this case. When plugged into the above formula, the result is =0.2718121285, or 27.18% probability.

Please post one question at a time and learn from the answer before you post the next question.

2)

I guess the easiest way to do this is check the Pythagorean theorem for a right triangle.

For each value of c find the length squared of each of the 3 sides of the triangle and sort them.

Then if the triangle is a right triangle the sum of the two smaller squared sides will equal

the largest squared side.

I'm going to let you slog through the arithmetic.  I suggest using a spreadsheet or something.

1)

\(-3x+4y=4 \\ y = \dfrac 3 4 x + 1 \\ slope = \dfrac 3 4 \\ \text{now we construct a line with slope }\dfrac 3 4 \text{ passing through }(4,0)\\ \text{using point slope form}\\ (y-0) = \dfrac 3 4 (x-4) \\ y = \dfrac 3 4 x - 3 \\ -3x + 4y = -12\)

\(\text{here may be some clever way to attack this but if so I'm not aware of it}\\ p(x) = a x^3 + b x^2 + c x + d\\ \text{using the function values give we can form the equations}\\ -a+b-c+d=15\\ d=0\\ a+b+c+d=5\\ 8a+4b+2c+d=12\\ \text{I would dump these into a matrix and solve it like that}\\ \begin{pmatrix}-1 &1 &-1 &1\\0 &0 &0 &1 \\1 &1 &1 &1 \\8 &4 &2 &1\end{pmatrix} \begin{pmatrix}a \\ b\\ c \\ d\end{pmatrix} = \begin{pmatrix}15 \\0 \\5 \\12 \end{pmatrix}\)

Solving this using the method of you choice  we find

\(a=-3,~b=10,~c=-2,~d=0 \\ p(x) = -3x^3 + 10x^2 -2x\)

The x-intercepts occur where p(x)=0, factoring we get

\(p(x)=-3x^3+10x^2-2x = \\ -x(3x^2 - 10x+2)\\ \text{and this has zeros at } x=0 \text{ and}\\ x = \dfrac{10\pm \sqrt{100-24}}{6} = \dfrac 1 3\left(5 \pm \sqrt{19}\right)\)

ok so basically the zeros of this are

\(x =\dfrac 1 2 e^{\dfrac{i(\pi + 2\pi k)}{4}},~k=0,1,2,3\)

and then what we do is pair up the conjugates and form

\(\left(x - \dfrac 1 2 e^{i\pi/4}\right)\left(x - \dfrac 1 2 e^{-i\pi/4}\right) \left(x - \dfrac 1 2 e^{3\pi/4}\right)\left(x - \dfrac 1 2 e^{-3i\pi/4}\right)\)

which becomes

\(\left(x^2 - \dfrac{x}{\sqrt{2}}+\dfrac 1 4\right)\left(x^2+\dfrac{x}{\sqrt{2}}+\dfrac{1}{4}\right)\)

Solution:

\( \small \text {Translate Joules to Liter-atmospheres}\\ w = -5000J * \dfrac{ 1L * atm} {101.325J} =-49.35L*atm \,| \small \text { Negative (-) means energy entered system.}\\ w = −P*\Delta V\\ \small \text {Change in volume }\\ \Delta V = -\dfrac {w}{P} = -\dfrac {-49.35L*atm }{1 atm}=+ 49.35L \,| \small \text { Positive (+) means gas expanded.}\\ \text { }\,\\ \text {Final Volume of the cylinder } 49.35L\)

GA

Hi Guest,

What do you think the answer might be?

One of your class or group members already asked this question

Here ya go:

Fixed monthly charge is the charge when yo send ZERO texts....    at text=0 the charge is  L 8

Leaman essentially lets you have 40 texts before the cost is the same as mcateer (L12)....after 40 texts the leaman plan becomes more costly.

I think that is going to be pretty ugly....Are you sure it isn't  16x^4 - 1   ?

Find a common DENOMINATOR

for  8 and 3 this would be 24

then convert the mixed numbers to improper fractions

19 1/8 = 19 3/24 = 459/24

14  2/3  =  14  16/24 = 352/24

Now subract them

459/24 - 352/24 =  (459-352)/24 = 107/24  

Now convert back to mixed number and simplify the fraction (if possible)

107/24 = 4 11/24          Done! 

Ths abuse that happens on this site also frustrates me and I'd like to see far less full answers.

Howerver, there are some students here that are definitely learning from the answerers expertise.

Should they be denied access to a site like this becasue others abuse what the site offers.

If I was still actively teaching in a classroom I have no idea how I would cope with sites like this one.

They would frustrate the h**l out of me.  I guess I would just have to accept that they facilitate learning and/or they facilitate cheating. KnockOut is right in that a students will always find ways to cheat. It is largely up to them how they use or abuse available resources.

But the fact that sites like this exists means that controlled exams are more vital than ever!

Interpreting Rom's answer:

If you ask a proper question you might get a proper answer.

Otherwise you get a bad joke, or worse.  :D