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I figured it was 10 solutions... thank you, CPhill! 

8^2 = 64       9^2 = 81        72 is in between

8.something is the answer probably around 8.4   

  try to multiply 8.4  x  8.4  (without a calculator)   then try 8.5 x 8.5 to see which is closest

another way:

sqrt72 = (sqrt 2 x 36) = 6 x sqrt2     I know sqrt 2 is approx  1.414     6 x 1.414   = ~ 8.5  (without a calculator)

sin 5x  = 1/2

Note that the "5" "horizontally compresses" the normal sine curve by a factor of 5

So......for sin (x), there will 2 solutions on (0, 2pi) where sin x = 1/2

So....2 values in one normal period

But....we have 5 periods  on (0, 2pi)  for sin (5x)

So.....there will be  5(2)  =  10  solutions  for  sin (5x)  = 1/2

1)   Explicit  

Number sold in any week, n  =   510(1.12)^(n - 1)

2)  Total phones sold in first year

510 [ 1 - 1.12^52 ] / [ 1 - 1.12 ] ≈ 1,536,478

√72  =  √2 *√36     √2 * 6  =  6√2

First balance the equation

                           C2H4---> 2C+ 2H₂

C₂H₄  +  6 F₂ ---------------------------------->  2 CF₄  +  4 HF   

                                                                    Note that we got 4 HF's   Given 2 HF's = - 537kj       - 537 x 2

                                                                                   we got 2 CF₄     Given 1 CF₄ = - 680 kj       - 680 x 2

                                                                                                                                                      ___________

                                                                                                                                        deltaH = - 2434kj

                                                                                            But I will assume the intermediate reaction occurs too:

                                                                                   C₂H₄ -----> 2C + 2H₂    we have ONE of these reactions  for  - 52.3 kj

        deltaH becomes - 2434 - 52.3 kj =  -2486.3 kj

sqrt [  (2--3)^2 + (-5-7)^2  ]

sqrt ( 5^2 + (-12)^2 )

sqrt (169)        or 13  

If I understand the question correctly, 48/100 reduces to 12/25. 

Actually, (n)(2pi), not (2n)(pi). After n periods, which is n(2pi), the value of the function is still the same.

You can derive this from a right triangle.

\(\displaystyle{ \sum_{r=k+1}^{60} (3r+2)\log_8\left(4^r\right)\\ =\sum_{r=k+1}^{60} \left(2r^2 + \dfrac{4}{3}r\right)\\ =2\sum_{r=k+1}^{60}r^2 + \dfrac{4}{3}\sum_{r=k+1}^{60} r\\ =2\left(73810 - \dfrac{k(k+1)(2k+1)}{6}\right)+\dfrac{4}{3}\left(1830-\dfrac{k(k+1)}{2}\right)\\ =150060-\dfrac{1}{3}\left(2k^3+3k^2+k\right)-\dfrac{2}{3}\left(k^2+k\right)\\ =150060-\dfrac{2}{3}k^3-\dfrac{5}{3}k^2-k\\ 150060-\dfrac{2}{3}k^3-\dfrac{5}{3}k^2-k > 106060\\ 2k^3+5k^2+3k-132000<0\\ k < 39.58 }\)

Greatest value of k is 39.

\(\log_8\left(4^r\right)\\ =r \log_8\left(4\right)\\ =\dfrac{2}{3} r\)

c.

GD = \(\dfrac{\text{BD}}{2} \approx 3.24\text m\) 

Entire height of the house is \(\text{GD} \tan\left(50^{\circ}\right) + 6\text m= 9.86\text m\)

b.

Construct line DP such that DP is perpendicular to HE and P is on line HE.

PE = \(\dfrac{3\text m}{\tan\left(50^{\circ}\right)} \approx 2.52\text m\)

BD = HE - PE = (9 - 2.52)m = 6.48m which is around 6.5 m.

a.

BH = 6m - 3m = 3m

DE = \(\dfrac{3\text m}{\sin\left(50^{\circ}\right)} \approx 3.92\text m\)

THANKS YOU  for not justt giving answer as that is pointess for me to accualy learn how tom . do .  this kind of problem in the future

\(3Ax-24=5x-9+x\\ (3A-6)x=15\\ x = \dfrac{5}{A-2}\\ \text{When }A = 2, \text{there are no solutions.}\)

ok thanks

Yes, it is 48/99.

OWNer is running a business and wants it to do well and make a nice profit for him/her....likely the THIRD choice is the one.

a) Start with Sin(S) = 2/4       and      Tan (Q) = 2/4        Are S and Q different?

b) a rhombus has 4 equal sides.....

c)  two right triangles added together = total area    area of right triangle = 1/2 L₁L₂       L = leg

Good Luck !  

Remember SOHCAHTOA for right triangles?

a) Cos (B) = 8/17       now do th einverse cos to find B

b) Sin 20 = ?/11          Can you do it now?

c)Tan GHJ = 10/3

d) Pythagorean theorem   DIagonal = sqrt (10^2 + 3^2)

e) draw line from S to bisect side 10    then  1/2  Diagonal is hypotenuse  and one leg is 5     then calculate the top angle....which is 1/2 of angle HSI

See what you find!  

4 friends , scores 90 - 100 with the ringer Angie scoring 80.  

  The average (MEAN)  score will be (all three of the 90-100 scores PLUS Angies score)    /  four

the mean (average) SCORE WILL GO down Angie after decides to go to the Master's with her score of 75 (assuming the other scores do not change)

The MEDIAN score is the score in the middle of the 4....since there are 4 scores, you would take the middle TWO scores and average them to get the median.....the median would not change unless the middle two scores change in the final round.......Angie's lowe score would not change the median (if she is still the lowest scorer)

Sandy starts with 5 marbles and Sam takes 2 away from here      5   take away 2   is  5-2     Sandy now has 5-2 marbles

Jeffrey gives 1 marble to Sandy     so Sandy now has    5-2  +1 marbles

5-2   +1 ==== 3+1 = 4 marbles.         Fortunately Sandy hasn't lost all of her marbles ! 

6. The numbers are growing exponentially so there is no sum

8. The numbers are shrinking exponentially so there is a sum, which is 9.

Based on this, can  you find the other two?

You are very welcome!

:P