# Tutor profile: Harley E.

## Questions

### Subject: Trigonometry

When f(x) = sin(x), how would the variants a, b, c, & d affect the result of the sound produced by each transformation af(bx + c) + d, of any given wave. E.g. How would the variant 'a' affect the resultant sound of af(x)? How would the variant 'b' affect the resultant sound of f(bx)? How would the variant 'c' affect the resultant sound of f(x+c)? How would the variant 'd' affect the resultant sound of f(x)+d? Please consider the graph and how these would affect it graphically.

The variant 'a' affects the entire function. If 'a' would be greater than 1, then we would consider the question to be "Increase it." In this context "it" would refer to the volume. Graphically this would increase the amplitude (stretch in the 'y' direction). 'b' affects the input of 'x', so this would be increasing how often 'x' would occur, increasing the frequency of the wave. Graphically this would squash in the x-direction (inversely proportional to the value of 'b'). Sonically, if 'b' is greater than 1, then the pitch would increase. 'c', once again, only affects the input of 'x'. Considering sin(x) = y, no matter what has been added to the value, must still equal 'y', so when c>0, the graph will be shifted negatively. E.g. sin(x+90) = cos(x). Sonically this creates a very strange and interesting 'clip'. Our ears are not used to the hearing sound that does not start from 0, so creates an unnatural and quite horrible popping sound. Finally, 'd' again affects the entire function after everything has happened. The wave itself remains the same and is not affected intrinsically, however, the entire graph is shifted positively in the 'y' direction. Sonically this creates a very strange manipulation indeed. Due to the sound being made up of positive and negative fluctuations, what 'd' would create is an uneven peak to trough ratio and this physically creates audible illusions that wouldn't neccessarily always be detectable my the human ear but essentially stretchs the soundwave into inaudible bass frequencies or treble frequencies depending on whether 'd' is greater or less than 0.

### Subject: Geometry

A friend and I are hungry and look up some deals for pizza. The local shop gives us the following: 18" = £10 12" = £5 Should we share the larger one? Or get our own small?

The measurement of 18" (18 inches) describes the diameter of the pizza, so I am instantly interested in how much pizza we will actually be getting (area). These pizzas were circular, so to calculate the area we will have to use the formula of pi x r x r (r = radius). Therefore for the larger; d = 18, r = 9 & when d = 12, r = 6. Just for the sake of understanding these values and ease of calculations, we can keep our answers in terms of pi. A = 81π square inches A = 36π square inches But I would have wanted a whole 12" to myself! That still would only be 72π square inches of pizza altogether, even when we do buy two for the same price as the singular 18". If the amount of pizza is most important to you, go for the 18" as you can split 81π square inches of pizza instead of only 72π worth of pizza. However! If it's the crusts you are more interested in, then we will have to find out the circumference value (C). π = C / d, therefore C = πd, a lot more of an easier calculation to make: 18π" shared between 2, will mean you may only get 9π" of crust! But if you have your own then you will get a full 12π"! So up to you: pizza, or crust. And that's just for circular pizzas! And we haven't even discussed whether you're interested in thin-crust vs deep pan!

### Subject: Algebra

A crate of beer holding only 18 cans of 500ml in size, each has an ABV of 4.8% but costs £25. Another crate has 20 bottles, but each with a vol. of 330ml, has an ABV of 5.0% but costs £20. Which is better for value and why?

First to work out how much liquid would be in each crate: 18 x 500 = 9L 20 x 330 = 6.6L Seeing ABV uses '%' as its unit of measure, let's calculate the hypothetical amount that would be pure alcohol: 9 x 0.048 = 432ml 6.6 x 0.05 = 330ml These are the volumes of pure alcohol that will be in each crate, based on each price, so how much would we get for each £?: 432 / 25 = 17.28%/£ 330 / 20 = 16.5%/£ You can see that I've left our units of measure as an ABV per 1£. This reinforces how we got to our answer and why. In the first crate, we will be getting 17.28ml of pure alcohol for every £1 spent. Yet in the second crate, we would only be receiving 16.5ml for every £. Crate A will be better for money!

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