Alan

The circumference of a circle is given by pi*diameter.  The diameter here is 40 in, hence the circumference is 40pi in.  270° indicates that the arc is 3/4 of the circumference, so the length of the arc is (3/4)*40pi in or 30pi in.

If you divide by a fraction that's the same as multiply by the inverse of the fraction

\(\frac{3}{10}/\frac{1}{3}\rightarrow \frac{3}{10}\times \frac{3}{1} \rightarrow \frac{9}{10} \rightarrow0.9\)

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You need to use brackets to avoid ambiguity.

(3k - 6)/3 = 5

Multiply both sides by 3

3k - 6 = 3*5  or   3k - 6 = 15

Add 6 to both sides

3k = 15 + 6   or  3k = 21

Divide both sides by 3

k = 21/3  or  k = 7

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If the following image represents the situation:

then the solution is as follows (where V = volume and S  surface area):

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This might help answer your question Chris:

It isn't defined for values of t = 0, -1, -2, ... etc though, where you get division by zero.

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If you are talking about the sum of the first n-1 positive integers then it is given by:

SUM(n-1) = (n-1)*n/2

so SUM(6-1) = (6-1)*6/2 → 5*3 → 15

(i.e.    1 + 2 + 3 + 4 + 5 = 15)

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It means you have to find 

sin(Q),  cos(Q),  tan(Q),  cosec(Q),  sec(Q),  cot(Q)

I interpret this question as:

What is the minimum number of pens that must be selected (blindly) such that Lucy is guaranteed to have 16 pens of the same colour in the selection?

It is possible (though perhaps unlikely) that she selects in the following order:

5 black pens

15 white pens

15 blue pens

16 red pens (or possibly 15 red pens followed by 1 blue pen for these last 16)

The total number she has to select in order to guarantee getting 16 of one colour is therefore 51.

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"4.5!  is the gamma function of 4.5"

Not quite.   Gamma(n) = (n-1)!

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Gamma(4.5) = 11.632   which is between 3! = 6 and 4! = 24

For a linearised (i.e. small amplitude), unforced pendulum the following is a state-space representation, where the state space is represented by angle and angular velocity:

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