All Answers 353121 Answers

Both acute amgles in a right triangle sum to 90°

So.......the other acute angle  is  [90 - 52]°

Can you finish this one, NSS ??

No inequality is shown....

0!   + 1*1! + 2*2!   + 3*3!   +  4*4!  + 5*5!  .......+ 50*50!

One of our members, Heureka, can probably prove this in a more elegant manner....but....here's my best attempt

Note the first few terms

1  + 1^2 *0! + 2^2 * 1!   =   6     = 3!     

1 + 1^2*0! + 2^2 * 1!  + 3^2*2!   =  24   =  4!

1 + 1^2*0! + 2^2 * 1!  + 3^2*2!  + 4^2* 3!     = 120    =  5!       and

1 + 1^2 *0! + 2^2 * 1!  + 3^2^2!  + 4^2* 3! + 5^2*4! = 6!

So....it would appear that we can write the sum as  

1   + 1*1! + 2*2!   + 3*3!   +  4*4!  + 5*5!  .......+ (n)^2 * (n - 1)!!   = (n + 1)!

So

0!   + 1*1! + 2*2!   + 3*3!   +  4*4!  + 5*5!  + .......+ 50*50!    =

1 + 1^2 *0! + 2^2 * 1!   + 3^2^2!  + 4^2* 3! + 5^2*4 +  ....+ 50^2 * 49! = ( 51) ! 

And largest prime that would divide this   is    47

Side RF  ...(the hypotenuse) is opposite angle E

We know the adjacent side to the angle.....and we want to find the hypotenuse [ the side opposite the right angle in the triangle ]

The cosine relates the ratio of the adj side to the hypotenuse

So....we have

cos 60    =   7 / hyp       multiply both sides by hyp

hyp * cos 60  = 7        divide both sides by cos 60

hyp   =   7/ cos 60   =   14  (ft)    =   the ladder length

Yep....periodic

If we modeled the levels by a continuous curve....we would have an  up/down/up/down  pattern

[ A curve that "rises" and "falls" from left to right and repeats this pattern, over and over ]

Tangent of an anle   = opposite side / adjacent side   

So

Tan S =  12 / 5

Here's my best attempt

Saving cash  is always good

Bond investments  can have known positive returns on an investment   [ unless there is a bond default ]

The stock market is a gamble.....no positive returns are guaranteed...but......one way to  mitigate losses is by diversifying your  stocks...[ thus....a loss in one stock might be lessened by a gain in another stock ]

That seems better! Thank you, CPhill! 

Azsun's answer is perfectly valid...but..because we have 10 raised to a power..using the base 10 log seems more natural

log 10^(z + 2)   = log 27      and we can write 

(z + 2)  log 10   = log 27          [log 10 = 1...so...we can ignore this ]

z + 2    =  log 27        subtract 2 from both sides

z = log 27 - 2  ≈  -0.569

There has to be! Maybe combinations!

Translating this, we get \(\ln \left(10^{z+2}\right)=\ln \left(27\right).\) Now, we just apply the log rule(\(\log _a\left(x^b\right)=b\cdot \log _a\left(x\right)\)) , and get  \(\left(z+2\right)\ln \left(10\right)=\ln \left(27\right).\)

Then, we can simplify this to be  \(\left(z+2\right)\ln \left(10\right)=3\ln \left(3\right).\)So, we solve it, and our final answer is \(z=\frac{3\ln \left(3\right)}{\ln \left(10\right)}-2.\)

Thanks....but.....I do want to point out that more people than just the  mods give good answers, too !!!

Is there some "formula" for computing this, heureka???

Thanks

Confirmed      12 dollars

Number the squares as

1     2      3

4     5      6

7     8      9

We have the following possible sets of three different numbers   [ assuming that each dart lands in a different square ]  =

C (9 , 3)   =  84

But.....there are only 8  different sets  hat will form vertical, horizontal or diagonal patterns

So....the probability  is     8 / 84   =   2 / 21

Sine of an angle  = the ratio of the opposite side to the hypotenuse

So....for angle R

Opposite side  = 5

Hypotenuse = 13

So

sin R   = 5 / 13

Here's the first one

Actually....angles  A and C are equal

And angles  E and C are equal

So....by AA conguency.....triange ACB is similar to triangle ECD

So....angle  BAC = angle DEC...and these are opposite imterior angles 

And angle CBA = angle EDC...and these are oppositie interior angles

And AE and BD are transversals cutting AB and DE

But...when transversals cutting two line segments make opposite interior angles equal....the segments are parallel

So   AB ll DE

Haha! Your explanation is still better! 

Heck.....tertre beat me to it.....LOL!!!

So.....let b be the base.....and we have

(1b + 4)  + ( 2b +4)    = 4b + 1      simplify

3b + 8   = 4b + 1         subtract 3b, 1 from both sides

b = 7

Check

14₇ + 24₇ =  41₇

In base 10

11 + 18  = 29     ⇔   True!!!

We first translate this into an equation:

\(1*b^1+4*b^0+2*b^1+4*b^0=4*b^1+1*b^0\).

This is equal to \(b+4+2b+4=4b+1\).

Solving we get, \(3b+8=4b+1, b=7\).

Thus, the answer is base \(\boxed{7}.\)

We know an angle and the adjacent side to this angle

And we are looking for the hypotenuse [ the side opposite the right angle ]

The cosine  = the ratio of the adjacent side to the hypotenuse....so

cos (20°) =   54 / x  ⇒ "C"