Additional Maths question help Cosine formula

I was doing this past paper:
http://www.ocr.org.uk/Images/130160-question-paper-unit-6993-additional-mathematics-advanced-level.pdf

For Q7, I keep getting 9.286km as the answer even after looking at the mark scheme and reproducing the same calculation to the letter. Can someone point out where I'm going wrong or show correct working?

Here's the mark scheme if needed:
http://www.ocr.org.uk/Images/135673-mark-scheme-unit-6963-additional-mathematics-june.pdf

The answer in the markscheme is correct. I didn't work it out the way they worked it out, but their way works perfectly fine and the calculation is correct.

Firstly, are you remembering to root the answer? Because the cosine rule gives you a^2, not a. However, 9.286 is wrong even unrooted. You must be entering it incorrectly into your calculator, or maybe it's set in radians or something?

I got 3.11 now after typing it into my calculator, I did it on the Google calculator before which was probably why. What are radians?, because I searched it up and I saw that the google calculator uses radians.

Radians are another way of measuring angles. The problem with degrees is that they don't have a lot of mathematical significance. One degree is defined as $\frac{1}{360}$ of the of the total angle around a point. There's nothing intrinsically wrong with doing this, but the fact remains that the number 360 is arbitrary. Degrees are useful in lower level mathematics because 360 has so many divisors, giving us very simple integer answers for things like the interior angles of triangles, squares, hexagons etc. Radians on the other hand have a lot more mathematical meaning. A radian is the angle at the centre of the circle subtended by the arc such that the radius of the circle is equal to the arc length. So 360 degrees turns into 2pi radians. This might sound slightly complicated, but because this number system is based on a mathematical identity rather than a random number, there are a number of pleasing benefits which stem from their use. For instance, it suddenly becomes a lot simpler to calculate a number of things. The length of an arc is now given by $r\theta$, the sector area is given by $\frac{1}{2}\theta r^2$ etc. There are also many more reasons why radians are useful in more advanced mathematics. For instance, if you use radians rather than degrees, the area under the sine curve between 0 and pi/2 radians is 1.

So basically, radians are what degrees ought to be.

I know this probably isn't the right place to ask but I asked before and no one answered. What's the difference between SD and Variance. I know variance is SD squared but why? Why use one instead of the other? Why are they two different things when one is just the other squared/square rooted? Which would I use to investigate variance? (Diesel car's prices are more varied than that of petrol car's).

By the way, is that hypothesis too similar to: Diesel cars have a larger distribution of prices than petrol cars.

A set of values has a mean. The variance is the average squared difference between all the values in that set and the mean (the difference is squared because the average unsquared difference will be 0). I'm sure the variance itself has uses, but I am not aware of them and there definitely aren't any direct uses in GCSE Stats or S1.

As you know, the standard deviation is the root of the variance. Why this is I don't know, and I can't claim to know an awful lot about the standard deviation, but there are a lot of very important uses. I'm not sure if you cover the normal distribution in GCSE statistics, but it's a very important distribution that can be used to model an extraordinary number of things - it is quite possibly the most important statistical distribution in existence. The standard deviation is very important in the normal distribution because it allows you to map any modified normal distribution onto the standard normal distribution. Additionally, you know that 68% of points will lie within one SD of the mean, 95% will lie within 2SD and 99.7% will lie within 3SD.

What hypothesis is too similar?

Are these two hypotheses similar?
The price of diesel cars will have a wider distribution than that of petrol cars (I'll use histograms and box plots for this)
The price of diesel cars will be more varied than that of petrol car's. (I'll use Standard Deviation for this)

Yes, we do learn normal distribution. By the way, why isn't it 100%. Why only up to 99.7% and why such awkward uneven intervals of 68%, 95% and 99.7%

As I've already said, I haven't taken GCSE Statistics so I don't know. But those two hypotheses sound more or less identical to me, so I definitely wouldn't take them both.

The equation of the normal distribution is very complicated. There definitely is a pattern behind those numbers, but I don't know the normal distribution well enough to give you an explanation other than "it is what it is".

However, the normal distribution never reaches 100%. In the normal distribution, it is theoretically possible to get any value. Obviously, it gets exceedingly less and less likely the more SDs you go from the mean (e.g. after 7 SDs the chance is 99.9999999997440%) but theoretically, any value can come up.

http://www.mathopenref.com/calculator.html

Probably a better calculator than Google

a^2=81+64-144 Cos(20)
a^2=9.68
a=3.11

Sorry if this has already been explained, but the amount of writing in thread is too much to read at this time

6 days ago... I really am stupid for bringing this back up... Well...