Mathematics Question Thread

[RuiAce]

Hi is there a process in order to derive the formulas for surface area or volume in an array of shapes  in case you forget them in the test!
And also could someone please provide a really simple explanation to the topic of prob. when do you add and when do you times, what do each represent?
Thanks. 

In general, for the non circular prisms and pyramids, for the surface area you can always just literally compute the area of each face by hand. For the volume, for the prism it is is always of the form \(V = Ah\), where \(A\) is the area of the base. For a pyramid, it is always 1/3 that of the volume of a prism.

The surface area of the cone and sphere are pretty hard to derive. For the cylinder, you can be careful, so long as you know that the circumference of a circle is \(2\pi r\) to figure out the area of the rectangle.
The volume of the cylinder and cone can be done the same way as above: basically just let \(A = \pi r^2\). The volume of the sphere is again pretty hard to derive.

[jamonwindeyer]

The volume of the sphere is again pretty hard to derive.

100% an example in one of my Tutesmart classes or exams

[RuiAce]

100% an example in one of my Tutesmart classes or exams

But like why \(\pi \int_{-r}^r \left(\sqrt{r^2-x^2}\right)^2 \mathrm{d}x \) when one can \( \int_0^{2\pi} \int_0^\pi \int_0^r \rho^2 \sin\phi \,\mathrm{d}\rho\mathrm{d}\phi\mathrm{d}\theta \)

[Mate2425]

Hey guys for Exponential growth and decay questions if they do not specify how many numbers to round to/sig fig to, would you be marked correct e.g in HSC 2014 - http://educationstandards.nsw.edu.au/wps/wcm/connect/aea5d47c-525d-4202-9d8c-72f76575a1fc/maths-hsc-mg-2014.pdf?MOD=AJPERES&CACHEID=ROOTWORKSPACE-aea5d47c-525d-4202-9d8c-72f76575a1fc-lGhCSoB
If you write answer as 2Kg ( nearest whole number) as A was originally given as 20kg?
Thanks 

[clovvy]

Hey guys for Exponential growth and decay questions if they do not specify how many numbers to round to/sig fig to, would you be marked correct e.g in HSC 2014 - http://educationstandards.nsw.edu.au/wps/wcm/connect/aea5d47c-525d-4202-9d8c-72f76575a1fc/maths-hsc-mg-2014.pdf?MOD=AJPERES&CACHEID=ROOTWORKSPACE-aea5d47c-525d-4202-9d8c-72f76575a1fc-lGhCSoB
If you write answer as 2Kg ( nearest whole number) as A was originally given as 20kg?
Thanks 

If they don't signify how many sig figs required, you just need to state it at the end of your answer I believe... Because my teacher told me that for HSC maths, what markers look for it the process you take to obtain your answer

[8veFable]

Hello!

I am struggling with Q16 ii) from the 2012 2U exam.

I tried to look at the solutions and manage to get that point Q is ([1 - sinθ]/cosθ) by using part i) to find the point of intersection with y = 1 but I am massively confuzzled as to how you are meant to use that information to find BQ.

Any help would be appreciated!

Edit: I did try and use the point intercept formula y - y1 = m(x - x1) but that didn't work out :O

[RuiAce]

Hello!

I am struggling with Q16 ii) from the 2012 2U exam.

I tried to look at the solutions and manage to get that point Q is ([1 - sinθ]/cosθ) by using part i) to find the point of intersection with y = 1 but I am massively confuzzled as to how you are meant to use that information to find BQ.

Any help would be appreciated!

Edit: I did try and use the point intercept formula y - y1 = m(x - x1) but that didn't work out :O

If you know that \(Q\) is the point \( \left( \frac{1-\sin\theta}{\cos\theta}, 1 \right) \), then by the distance formula the length of \(BQ\) is just \( \sqrt{\left( \frac{1-\sin\theta}{\cos\theta} - 0\right)^2 + (1-1)^2} \), and note that the 1 - 1 cancels out to 0.

(Alternatively, it's just the difference between their \(x\)-coordinates, because \(BQ\) is a horizontal line segment. And \(B\) obviously has \(x\)-coordinate 0 since it lies on the \(y\)-axis.)

[S200]

Hello!

I am struggling with Q16 ii) from the 2012 2U exam.

I tried to look at the solutions and manage to get that point Q is ([1 - sinθ]/cosθ) by using part i) to find the point of intersection with y = 1 but I am massively confuzzled as to how you are meant to use that information to find BQ.

Any help would be appreciated!

Edit: I did try and use the point intercept formula y - y1 = m(x - x1) but that didn't work out :O

Hmm. I'm missing something here I think, but yeah.

The only way I can think of doing it is to find the length of PO using \(\sin{\theta}\) and the length PT.
Then make a right angled triangle with Q, a point vertically down from Q (say, Y), and P.
You can use tan to find length PY and minus that from PO to give you BQ.

The angle QPY would be \(\frac{-1}{\tan {\theta}}\)

If you know that \(Q\) is the point \( \left( \frac{1-\sin\theta}{\cos\theta}, 1 \right) \), then by the distance formula the length of \(BQ\) is just \( \sqrt{\left( \frac{1-\sin\theta}{\cos\theta} - 0\right)^2 + (1-1)^2} \), and note that the 1 - 1 cancels out to 0.

(Alternatively, it's just the difference between their \(x\)-coordinates, because \(BQ\) is a horizontal line segment. And \(B\) obviously has \(x\)-coordinate 0 since it lies on the \(y\)-axis.)

Edit - Rui's way is a lot easier.

[terassy]

Help for part(ii). This is from HSC 2U Maths 2007


Part (ii) states 'during which month and year'. So does that mean the month it will reach 4000 million (i.e. October) or like the month it will enter, with more than 4000 million (i.e. November).

Thanks

Final answer to part (i)

help ;(

[jamonwindeyer]

Help for part(ii). This is from HSC 2U Maths 2007
(Image removed from quote.)

Part (ii) states 'during which month and year'. So does that mean the month it will reach 4000 million (i.e. October) or like the month it will enter, with more than 4000 million (i.e. November).

Thanks

Final answer to part (i)

help ;(

Hey! I read it as during the month, so October should be the answer

[8veFable]

Hi! Thank you Rui and S200 for answering my question! After your responses I realised the question asked for length' instead of finding the equation, so that made me much less confused 

I also have another question if you anyone could please help me with it.

I don't know how on earth I forgot this all of a sudden but could someone please clarify if calculus of trigonometric functions is done in degrees or radians? Like for example if you were given a motion question and it told you to sub in 2 seconds (At t = 2), then would the calculator be set in degrees or radians?

Is using the wrong mode a mistake that is usually done in motion questions with trigonometric functions?

Much appreciated!

[RuiAce]

Hi! Thank you Rui and S200 for answering my question! After your responses I realised the question asked for length' instead of finding the equation, so that made me much less confused 

I also have another question if you anyone could please help me with it.

I don't know how on earth I forgot this all of a sudden but could someone please clarify if calculus of trigonometric functions is done in degrees or radians? Like for example if you were given a motion question and it told you to sub in 2 seconds (At t = 2), then would the calculator be set in degrees or radians?

Is using the wrong mode a mistake that is usually done in motion questions with trigonometric functions?

Much appreciated!

Always radians with the calculus of trig functions.

You should never use degrees unless you explicitly see the little circle hinting at degrees

[markkhodair]

Hey! I know that this question has probably been asked and answered so many times in the past, so I apologise for repeating, but I just wanted to clarify when to use radians mode in the calculator as opposed to using degrees mode? Thank you!

[fun_jirachi]

You should never use degrees unless you explicitly see the little circle hinting at degrees

Pretty much this.

[RuiAce]

Hey! I know that this question has probably been asked and answered so many times in the past, so I apologise for repeating, but I just wanted to clarify when to use radians mode in the calculator as opposed to using degrees mode? Thank you!

Just watch out for any posts that are made on the same page as well in the future. Your question was answered in the post immediately above yours.
(That rule of thumb (as fun_jirachi bumped) applies not just to calculus, but in any generic contexts as well.)