Find cot θ if csc θ = negative square root of thirty seven divided by six and tan θ > 0.

First of all, let us write down what we know:

a) 1 pint in Britain costs 65p

b) 1/3 litre in France costs 4 francs

c) 1 pint is approximately 0.568 litres

Now, our goal is to find the cost of a pint of beer bought in France. For this question, there are two main ways that I can think of for solving it so I will show both of these below:

1) This method relies on finding a conversion factor between 1/3 of a litre and 0.568 litres (ie. by how much must we multiply 1/3 of a litre to attain 0.568 litres, or you can think of it as how many 1/3s of a litre are there in one pint) and applying it to the price in France (ie. we would multiply the price of 1/3 of a litre in France by that value to obtain the price of 0.568 litres, or one pint, in France).

Thus, finding the conversion factor (let us call it x) stated above, we get:

(1/3)x = 0.568

x = 0.568 / (1/3)

x = 0.568*3

x = 1.704

Applying this to the price of 1/3 of a litre in France (which is 4 francs), we get:

Price of 1 pint* in France = 4*1.704

Price of 1 pint* in France = 6.82 francs (to two decimal places) which is approximately 7 francs

*Note here that we are calculating the price of 0.568 litres, which is 1 pint. Effectively, the price of a pint is the price of 1/3 of a litre multiplied by how many 1/3s of a litre there are in one pint, which is 1.704

Thus, the cost of a pint of beer bought in France is 7 francs (rounded to the nearest franc - not sure if there is any rounding specified in the question).

2) The second method is very similar, except that we are finding how many pints there are in 1/3 of a litre (ie. by dividing 1/3 of a litre by 0.568). Thus, with y denoting the number of pints, we get:

y = 1/3 / 0.568

y = 0.586854 (to 6 decimal places)

Note above that I haven't rounded to two decimal places, for example, because we still require this value for future calculations and by rounding too much we can skew the value of the answer slightly - it is always best practice to use exact values however.

Now if there are (approximately) 0.586854 pints within 1/3 of a litre, to find the price of 1 pint, we need to divide the price of 1/3 of a litre of beer in France by 0.586854. Thus, we get:

Price of 1 pint in France = 4/0.586854

Price of 1 pint in France = 6.82 francs (to two decimal places), which is approximately 7 francs

Thus, the price of one pint of beer in France is 7 francs (rounded to the nearest whole franc).

The other point I wasn't entirely sure of was whether you wanted the cost in British or French currency.

Apart from that, the key point to remember is that you are finding ways to establish relationships between different forms of measurement, and applying them.

I find it helpful to do this sometimes with simpler examples for numbers given, for example for this question you could have said: if one pint is 2/3 of a litre, and 1/3 of a litre of beer in France costs 5 francs, how much would one pint of beer cost in France? In that case, you would naturally say that a pint is twice 1/3 of a litre, therefor it would cost 10 francs.

What you have actually done is probably:

(2/3) / (1/3) = 2

5*2 = 10

Notice that this is exactly what we did in the first method, just with slightly different numbers. Thus I would advise you to not get overwhelmed by fractions or decimals but to work through the question step by step.

I hope this made sense, however if you have any further questions or comments, feel free to ask below :)