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No prob! Thanks! 

Oh yeah! Thank you!!! I really appreciate your help! :)

LCM denominator is 8, since all denominators are are factors of 8. 

Convert the fractions to fraction over 8. 

6/8+24/8+4/8+8/8+3/8+8/8+6/8

=59/8 <-- improper

=7 3/8 <-- mixed 

=7.375

The diameter of the largest ball that can fit in a cube matches the length of the cube's edge.

volume of a sphere = \(\frac43 * \pi * \text{radius}^3\)

a.

diameter = 1 , radius = 1/2

volume = \(\frac43*\pi*(\frac12)^3=\frac{\pi}6 \approx 0.524 \text{ cubic feet}\)

b.

diameter = 2 , radius = 1

volume = \(\frac43 * \pi * 1^3 = \frac{4\pi}{3} \approx 4.189 \text{ cubic feet}\)

c.

diameter = 3 , radius = 3/2

Do you see what to do now?

If you need more help please just ask :)

You do realise that this was my riddle at first, did you get it from me? 

hi , how are you

But what if actually the man book a room for 2 people (with his friend) and decided to knock first before entering?

150  + 50N  =  1000      where N is the number of weeks we are seeking

Subtract 150  from both sides

50N   =  850       divide both sides by 50

N  =  17  weeks

Decimal to Fraction: Step 1: Write down the decimal divided by 1, like this: decimal 1. Step 2: Multiply both top and bottom by 10 for every number after the decimal point. (For example, if there are two numbers after the decimal point, then use 100, if there are three then use 1000, etc.) Step 3: Simplify (or reduce) the fraction. 

Decimal to Percent: To convert a decimal to a percent, multiply the decimal by 100, then add on the % symbol. An easy way to multiply a decimal by 100 is to move the decimal point two places to the right. This is done in the example below. Each decimal in Example 1 went out two places to the right of the decimal point.

AH, I know you would have seen that before long..

What's really ingenious is how you figured out that MrSilvers was trying to take the sin of 15º ...That took some real insight ! 

Very ingenious, hectictar.....it would not have occurred to me  to apply the half-angle formula!!!!

y"  +  25y   =  3sin2x

First....let's find the solution to the homogenous equation 

y" + 25y  = 0

The characteristic  equation  for this is

r^2  +  25   =  0     →    r   = ± 5i

So, the complementary  solution to this homogenous equation is 

y  =  c1 cos(5x)  +  c2 sin (5x)      (1)

Now let's solve the non-homogenous equation

y"  +  25y   =  3sin2x

Guess that the solution  is

y  = Acos 2x  + Bsin2x

y' =  -2Asin2x  + 2Bcos2x

y"  =  -4Acos2x -4Bsin2x

So  we have

-4Acos2x - 4Bsin2x  + 25(Acos2x + Bsin2x)  =  3sin2x

(25A -4A)cos2x   + (25B - 4B)sin2x  =  3sin2x

21Acos2x   +  21Bsin2x  =  3sin2x

This implies that A  = 0       and  B  =   1/7

So......the solution to  this is that

y  =  (1/7)sin2x    (2)

Combining (1)  and (2)  we have this

y  =   c1 cos(5x)  +  c2 sin (5x)   + (1/7)sin(2x)

y'  = - 5c1 sin (5x) + 5c2 cos(5x) + (2/7)cos(2x)

Applying the initial conditions

y(0)  =  c1 (1) = 0    →  c1  = 0 

y' (pi)  = 1

5c2 (-1)  + (2/7)(1)  = 1

-5c2 + 2/7  = 1

-5c2 =  5/7

c2  =  -1/7

So.... the solution with the applied intial conditions is

 y  =  (-1/7)sin(5x) + (1/7) sin(2x)

And

y' =  (-5/7)cos(5x)  + (2/7)cos(2x)

Check

y(0)  = 0 ??

(-1/7)(0)  + (1/7)(0)  = 0

y'(pi)   =  1   ??

(-5/7)(-1)  + (2/7)(1)

5/7  +  2/7   =  1

Accessory meaning: a thing that can be added to something else in order to make it more useful, versatile, or attractive. 

The first electronic device was the relay, a remote switch controlled by electricity. It made modern telegraphy possible and evolved into the repeater- thus the relay- a remote controlled switch, was in effect the first ( Electronic) device, though not anything involving crystals, diodes, vacuum tubes etc. The first invented Relay was used as part of his telegraph system circa 1844. 

https://en.wikipedia.org/wiki/History_of_technology for the first two questions of yours

853

___

999

(Interesting trick I learned: put a number over 9's repeating, and you get that number repeating, as long as that number has the same amount of digits.)

Guest, you are lucky one of the mods hasn't targeted you for this yet. But next time, please make the title of the post relevant to the post contents.

To answer the question in the title:

= 3.93464080752965375754 * 10¹⁴⁴

13 1/2.

Hey...that's a really good way of explaining it ! 

to find the sign of a number, you divide that number by its absolute value. If the number is 0, the sign is 0.

If you want to find the sine of an angle, you can draw a circle with radius 1. Then, starting at the top, go that many units around the circle. The x value is the sine, the y value is the cosine, and the x value divided by the y value is the tangent.

it is 5*sqrt(3)

Approximately 8.660254037844387

\(\cos^2(\frac{\pi}{8})+\cos^2(\frac{3\pi}{8})+\cos^2(\frac{5\pi}{8})+\cos^2(\frac{7\pi}{8}) \\~\\ \cos^2(\frac12\cdot\frac{\pi}{4})+\cos^2(\frac12\cdot\frac{3\pi}{4})+\cos^2(\frac12\cdot\frac{5\pi}{4})+\cos^2(\frac12\cdot\frac{7\pi}{4})\)

Apply the half-angle formula: \( \cos^2(\frac12\cdot a)=\frac12(1+\cos a) \)

\(\frac12(1+\cos\frac{\pi}{4})+\frac12(1+\cos \frac{3\pi}{4})+\frac12(1+\cos\frac{5\pi}{4})+\frac12(1+\cos\frac{7\pi}{4})\)

Now we can evaluate the cosines.

\(\frac12(1+\frac{\sqrt2}{2})+\frac12(1-\frac{\sqrt2}{2})+\frac12(1-\frac{\sqrt2}{2})+\frac12(1+\frac{\sqrt2}{2})\)

Simplify.

\((1+\frac{\sqrt2}{2})+(1-\frac{\sqrt2}{2}) \\~\\ 1+\frac{\sqrt2}{2}+1-\frac{\sqrt2}{2} \\~\\ 2\)

If you want more steps shown please just say so 

I'm feeling sharp?