Alan

Yes.  Click the 2nd button and you will see mod appear.

Possibly; possibly not!  You'll need to ask a specific abstract algebra question to find out.

The steradian is a measure of solid angle, whereas the degree is a measure of planar angle.  The steradian is 3d, the degree is 2d, so one cannot sensibly say there are so many degrees in a steradian.

Let the number of people in the first hall be  f  and the number in the second hall be  s.

Then we have

f + s = 200      ...(1)

5f/6 = s + f/6  ...(2)    (Because there are 5f/6 people left in the first hall)

From (1) we have s = 200 - f   ...(3)

Put this into (2) to get   5f/6 = 200 - f + f/6  or:   5f/6 = 200 - 5f/6   or  10f/6 = 200 so  f = 200*6/10 = 120

f = 120

Put this into (3) to get s = 200 - 120 or s = 80

Now see if you can do 1b and 1c yourself.  Come back if you still struggle.

This might refer to

(1) a 45°/45° right-triangle, with sides in the ratio 1:1:√2 so 

     sin(45°) = 1/√2,  cos(45°) = 1/√2,  tan(45°) = 1

(2) a 30°/60° right-triangle with sides in the ratio 1:√3:2 so

     sin(30°) = 1/2,  cos(30°) = √3/2, tan(30°) = 1/√3

     sin(60°) = √3/2, cos(60°) = 1/2,  tan(60°) = √3

The square floor must have sides of 6 inches.  The depth of the folded sides of the box must be (10 - 6)/2 = 2 inches.

Beaten to it by Chris this time!!

A property of logarithms is that log(x*y) = log(x) + log(y)

Another is that  log(x^-1) = -log(x)

So:  log(2*a*b) = log(2) + log(a) + log(b)

       log(5/a) = log(5) + log(1/a) = log(5) - log(a)    (as 1/a = a^-1)

      log(10*b) = log(10) + log(b)

Therefore: log(2*a*b) + log(5/a) - log(10*b) =  log(2) + log(a) + log(b) + log(5) - log(a) - log(10) - log(b)

                                                                      = log(2) + log(5) - log(10) 

                                                                      = log(10) - log(10)

                                                                      = 0

I see Melody beat me to it (with a slightly different approach)!

$$\frac{1.9891\times 10^{30}}{1.9008\times 10^{27}}=\frac{1.9891\times 10^{30-27}}{1.9008}=\frac{1.9891\times 10^3}{1.9008}=\frac{1989.1}{1.9008}$$

$${\frac{{\mathtt{1\,989.1}}}{{\mathtt{1.900\: \!8}}}} = {\mathtt{1\,046.454\: \!124\: \!579\: \!124\: \!579\: \!1}}$$

Let h be the height of the pole and d be the distance from the building.

tan(28) = h/d   ...(1)

tan(14) = (h-60)/d  ...(2)

Rewrite (2) as tan(14) = h/d - 60/d   ...(3)

Use (1) in (3):     tan(14) = tan(28) - 60/d  ...(4)

Rearrange (4) to get d = 60/(tan(28)-tan(14))

$${\mathtt{d}} = {\frac{{\mathtt{60}}}{\left(\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{tan}}{\left({\mathtt{28}}^\circ\right)}{\mathtt{\,-\,}}\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{tan}}{\left({\mathtt{14}}^\circ\right)}\right)}} \Rightarrow {\mathtt{d}} = {\mathtt{212.478\: \!562\: \!245\: \!221\: \!509\: \!7}}$$

or d ≈ 212.5 ft

There are 7 numbers which, when raised to the power of 7 result in -1.42.  However, all but one of them are complex.  Melody has identified the only real number solution.  Here are all the solutions: