New Maths Problem YT Channel

Hello all, hope you have all had a great Christmas and looking forward to the new year (and mocks/exams!)

I finished my A-Levels in 2015 and went on to university to study engineering and now work as an engineer in industry. However I have always enjoyed teaching/tutoring and thought about going in to teaching. To satisfy this itch I have created a maths(+science) Youtube channel called 'Solutions Explains'.

The idea is to essentially provide solutions to GCSE/A-Level/similar maths/science problems, but in a more visually attractive way that engages with students and a wider audience more effectively than simply posting a model answer.

I currently have 1 video up which is relevant to AQA A-Level maths - the question boils down to:

Find the maximum area of a rectangle inside a circle of radius 4.
Link to solution: https://youtu.be/NIjGmBXCkEM

I thought it would be worth making a post here on TSR to let people know about the channel as I always found this place very helpful and so wanted to give back to the community!

Let me know if any of you find this helpful or have some interesting suggestions or ideas for the future! I should have a new video up tonight too (GCSE level but still challenging!).

Thanks!

Reply 1

burntcherrio

12

Original post by Hilton184

Hello all, hope you have all had a great Christmas and looking forward to the new year (and mocks/exams!)

I finished my A-Levels in 2015 and went on to university to study engineering and now work as an engineer in industry. However I have always enjoyed teaching/tutoring and thought about going in to teaching. To satisfy this itch I have created a maths(+science) Youtube channel called 'Solutions Explains'.

The idea is to essentially provide solutions to GCSE/A-Level/similar maths/science problems, but in a more visually attractive way that engages with students and a wider audience more effectively than simply posting a model answer.

I currently have 1 video up which is relevant to AQA A-Level maths - the question boils down to:

Find the maximum area of a rectangle inside a circle of radius 4.
Link to solution: https://youtu.be/NIjGmBXCkEM

I thought it would be worth making a post here on TSR to let people know about the channel as I always found this place very helpful and so wanted to give back to the community!

Let me know if any of you find this helpful or have some interesting suggestions or ideas for the future! I should have a new video up tonight too (GCSE level but still challenging!).

Thanks!

yo, i had a go at the problem and came up with an alternate solution:
since the radius of the circle is 4, the distance from one edge to the center to the opposite edge should be 8. since its a rectangle we know the angles in a triangle made by its diagonal(8) will be 45 , 45 and 90. now that we know the hypotenuse of one such triangle and the angles, the side should be 8cos45 and 8 sin45, we multiply these 2 sides together to get the area 32 the same answer. so was my method a fluke and i got lucky or is it a legit solution( conceptually).
thanks, i enjoyed your method too, ( a possible idea for a video could be showing us different ways to reach a solution ie. algebraic, graphical or geometric)

(edited 4 years ago)

Reply 2

Hilton184

OP

9

Original post by burntcherrio

yo, i had a go at the problem and came up with an alternate solution:
since the radius of the circle is 4, the distance from one edge to the center to the opposite edge should be 8. since its a rectangle we know the angles in a triangle made by its diagonal(8) will be 45 , 45 and 90. know that we know the hypotenuse of one such triangle and the angles, the side should be 8cos45 and 8 sin45, we multiply these 2 sides together to get the area 32 the same answer. so was my method a fluke and i got lucky or is it a legit solution( conceptually).
thanks, i enjoyed your method too, ( a possible idea for a video could be showing us different ways to reach a solution ie. algebraic, graphical or geometric)

Hi, this is great, thanks for your response - I'm glad you enjoyed the video! And well done for trying it another way. It is possible to do through trigonometry, but you can not assume that the angle will be 45 degrees when you make the diagonal through the rectangle. The diagonal through the rectangle will only bisect the angle if the rectangle has all sides the same length (i.e. it is a square). And so you are assuming the rectangle is a square by your calculation and not proving is will be a square.

Instead, we can do this to prove it is a square:

Insert the diagonal through the rectangle, denoted k. The diagonal disects the angle into two, so the angle between the width and diagonal will be denoted 'z' and the angle between the length and diagonal will be denoted (90-z).

Now the width of the rectangle will be given by k*cosz and the length will be given by k*sinz.

Therefore the area of the rectangle will be length*width = k*cos(z)*k*sin(z)=k^2 *coz(z)*sin(z).

Using the double angle formulae, this becomes k^2*sin(2z).

Now we would need to find when sin(2z) is a maximum, which is when sin(2z) is equal to 1, which is when z equals 45 -- indicating that the rectangle must actually be a square -- and then your calculation to calculate the area of the square can be performed.

I might make a follow up video using this method as it is quite interesting actually!