Melody

Thanks Chris,

I did not see the bit about there being exactly 20 questions.    

3T+11M=100

So if could be

2 Multiple choice and 26 T/F      check     2*11+26*3 = 100

5 Multiple choice and 15 T/F      check      5*11+15*3 = 100

8 Multiple choice and 4 T/F        check       8*11+4*3 =   100

Incidentally this is a diophantine equation because all solutions must be integers.

There is a more formal way of solving so that all solutions are found.  (including negative solutions)

Of course you problems does not have any negative solutions :))

That is a LOT of digits.

 I do not think that I can count that high!!

Geno's answer is good but I will add that 

arc tan and inverse tan are the same thing.

On the web2calc you use atan 

but on many calcs the button looks like this     \(tan^{-1}\)

Hi Rom and guest,

I agree this is an interesting question but I would do it differently, Maybe my way is wrong.

Rom I don't think that you found the angle did you?

The staircase goes around the silo much like the label goes around a can of peas

If you take the label off the peas, it is a rectangle.  

The length will be the circumference of the can 2*pi*r

and the height will be the hieght of the can.

so

getting back to the silo, the stairs go from the bottom corner of the 'rectangle' to the diagonally opposite corner.

like this.

I wanted to draw the diagram but Geogebra has stopped working for me    

Anyway it is a rectangle.  and the stairs are the diagonal.

The rectangle is 37 foot hight and    2*pi*7 = 14pi       foot long

The incline angle is the angle between the base and the diagonal.

\(tan\theta = \frac{37}{14\pi}\\ \theta = atan\left( \frac{37}{14\pi} \right)\\ \theta = 40^04'\)

So the angle of incline is 40.072 degrees (3dp)

or 40 degrees and 4 minutes to the nearest minute

how do i factor 8x^3 -27x^2 -6x -8

\(8x^3-27x^2-6x-8\\ \)

This does not factorise nicely.  (Only oine of the roots is even real)

Perhaps you copied it wrongly?

Rom is correct of course but it also discriminates between other tytpes of roots as well.

The symbol for the discriminant is a triangle.

so

\(if \;\;ax^2+bx+c=0\qquad then\qquad x=\frac{-b\pm\sqrt{b^2-4ac}}{2}\\ The \;discriminant \;is\;b^2-4ac\\ \triangle=b^2-4ac \)

IF the discriminant = zero there will be one real root.   (sometimes called a double root)

IF the discriminant > 0 then there will be 2 real roots.

If the discriminant < 0 there there will be no real roots.  If you have not done complex numbers yet then you would probably say that there were NO roots.  This is because you cannot take the square root of a negative number.

ROM already said all this.

BUT also

If the descriminant is a perfect square then the roots will be rational 

If the descriminant is NOT a perfect square then the roots will be irrational 

38 followed by 22 zeros

supose that you had two rings that are the same size but one was 10K gold and the other was 18K gold which one would feel heavier?

I suppose it would depend on the density of the other elements in the two rings.

Gold is very dense so I would expect the 18K one to be heavier but if the other one contains lead  (which is heavier than gold)  then it would be heavier.

\((343^{1/3})^2=49\\ that \;means\\ (\sqrt[3]{343})^2=49\\ and also\\ \sqrt[3]{343^2}=49\)

Maybe this is related ???