Login

Welcome, Guest. Please login or register.

March 28, 2024, 06:57:46 pm

Author Topic: Mathematics Question Thread  (Read 1296555 times)  Share 

0 Members and 1 Guest are viewing this topic.

Calley123

  • Forum Regular
  • **
  • Posts: 71
  • Respect: 0
Re: Mathematics Question Thread
« Reply #3300 on: February 22, 2018, 07:28:24 pm »
0
Hey,

How do I find the shaded area of a parabola with x^2=4ay with a focus (0,5)  and a line y=a ? ( answer 8a^2/3) )
Also how do volume when it is rotated around y-axis ?  (Answer: 2 times pi times a^3 )
 Thanks !!

gilliesb18

  • Trendsetter
  • **
  • Posts: 123
  • Respect: 0
Re: Mathematics Question Thread
« Reply #3301 on: February 22, 2018, 08:03:56 pm »
0
Hello:)
I'm needing help(again) on an integration one...
this time its the sum of areas etc...
Qu. Find the area bounded by the curve y= 9-x^2 and the line y=5

Thanks so much ;D

jamonwindeyer

  • Honorary Moderator
  • Great Wonder of ATAR Notes
  • *******
  • Posts: 10150
  • The lurker from the north.
  • Respect: +3108
Re: Mathematics Question Thread
« Reply #3302 on: February 22, 2018, 08:43:47 pm »
+2
Hello:)
I'm needing help(again) on an integration one...
this time its the sum of areas etc...
Qu. Find the area bounded by the curve y= 9-x^2 and the line y=5

Thanks so much ;D

Hey! So you must draw a sketch for stuff like this, it makes things loads easier! If you sketch the parabola and the line, you'll see that the area we need is enclosed by the parabola on top, and the line on the bottom.

Remember, whenever we have an area between two functions, we just integrate the top curve minus the bottom curve:



But what are the limits? They tell us where the area starts and stops - We need the points of intersection of the parabola and the line. Equate them:



So, we need to evaluate the integral:



Reckon you might be able to take it from there? ;D this is a super important question to understand btw so if anything is confusing, ask! :)
« Last Edit: February 23, 2018, 10:15:36 pm by RuiAce »

jamonwindeyer

  • Honorary Moderator
  • Great Wonder of ATAR Notes
  • *******
  • Posts: 10150
  • The lurker from the north.
  • Respect: +3108
Re: Mathematics Question Thread
« Reply #3303 on: February 22, 2018, 08:48:55 pm »
+2
Hey,

How do I find the shaded area of a parabola with x^2=4ay with a focus (0,5)  and a line y=a ? ( answer 8a^2/3) )
Also how do volume when it is rotated around y-axis ?  (Answer: 2 times pi times a^3 )
 Thanks !!

Hey! Do you mean the area enclosed between the two? Take a look at the post I've just made above, because it's the same logic!! In your case we have an unknown \(a\), but it's the same method, integrate top minus bottom. In your case, the line is on top! Rearrange the parabola equation to make \(y\) the subject and:



The limits for you are where the parabola cuts the line, same as above!



So those are your limits, you're evaluating:



Edit: Actually this question is a tad confusing, because we can evaluate \(a\) using the known focus, \(a=5\), so I'm not sure why \(a\) even appears in the answer!
« Last Edit: February 22, 2018, 08:50:32 pm by jamonwindeyer »

Calley123

  • Forum Regular
  • **
  • Posts: 71
  • Respect: 0
Re: Mathematics Question Thread
« Reply #3304 on: February 23, 2018, 07:12:15 am »
0
Hey! Do you mean the area enclosed between the two? Take a look at the post I've just made above, because it's the same logic!! In your case we have an unknown \(a\), but it's the same method, integrate top minus bottom. In your case, the line is on top! Rearrange the parabola equation to make \(y\) the subject and:



The limits for you are where the parabola cuts the line, same as above!



So those are your limits, you're evaluating:



Edit: Actually this question is a tad confusing, because we can evaluate \(a\) using the known focus, \(a=5\), so I'm not sure why \(a\) even appears in the answer!


Thank youuu :)

slinkybench

  • Adventurer
  • *
  • Posts: 5
  • Respect: 0
Re: Mathematics Question Thread
« Reply #3305 on: February 23, 2018, 05:44:48 pm »
0
I need some help with this factorisation question:
x^4 - x^2 - 2x -1


arii

  • Forum Regular
  • **
  • Posts: 65
  • Class of 2018 Seniors
  • Respect: +8
Re: Mathematics Question Thread
« Reply #3306 on: February 23, 2018, 06:41:47 pm »
+7
I need some help with this factorisation question:
x^4 - x^2 - 2x -1



Hey there,

Please refer to the attachment for the solutions. I'm pretty sure that's all you can factorise (unless you are an Extension 2 Mathematics student).

Let me know if you don't understand anything.
2018 HSC | 4U Mathematics | 3U Mathematics | Advanced English | Chemistry | Physics | Legal Studies

Constantly getting my 4U Mathematics life saved on this website.

gilliesb18

  • Trendsetter
  • **
  • Posts: 123
  • Respect: 0
Re: Mathematics Question Thread
« Reply #3307 on: February 26, 2018, 06:22:22 pm »
0
Hey! So you must draw a sketch for stuff like this, it makes things loads easier! If you sketch the parabola and the line, you'll see that the area we need is enclosed by the parabola on top, and the line on the bottom.

Remember, whenever we have an area between two functions, we just integrate the top curve minus the bottom curve:



But what are the limits? They tell us where the area starts and stops - We need the points of intersection of the parabola and the line. Equate them:



So, we need to evaluate the integral:



Reckon you might be able to take it from there? ;D this is a super important question to understand btw so if anything is confusing, ask! :)

Ok great thanks heaps!!!

mirakhiralla

  • Trailblazer
  • *
  • Posts: 36
  • Respect: 0
Re: Mathematics Question Thread
« Reply #3308 on: March 02, 2018, 08:07:33 pm »
0
Hey! (:
I have attempted this question but I dont have answers and I am pretty sure it is not correct, if it isn't, can you please show me how to do it?

Terry borrowed $20000 on 1 January 2008. He agreed that on 1 January in each succeeding year he would pay back $3000 and add 6%p.a. interest on the amount owing during the year just completed. Find:
a) the amount still owing after ! January 2013
b) the number of years needed to pay off the debt

My answers: a) $22181.51  b) 120.7 years

jamonwindeyer

  • Honorary Moderator
  • Great Wonder of ATAR Notes
  • *******
  • Posts: 10150
  • The lurker from the north.
  • Respect: +3108
Re: Mathematics Question Thread
« Reply #3309 on: March 03, 2018, 12:24:38 am »
+2
Hey! (:
I have attempted this question but I dont have answers and I am pretty sure it is not correct, if it isn't, can you please show me how to do it?

Terry borrowed $20000 on 1 January 2008. He agreed that on 1 January in each succeeding year he would pay back $3000 and add 6%p.a. interest on the amount owing during the year just completed. Find:
a) the amount still owing after ! January 2013
b) the number of years needed to pay off the debt

My answers: a) $22181.51  b) 120.7 years

Hey! Yeah not quite, but that's all good, let me show you!

So we start with $20,000, let that be \(A_0\). After 1 year (so, January 2009), we add 6% interest and then pay back $3000. That looks like this:



The next year, we take that amount, and do the same thing. Add 6%, subtract $3000:



If you expand, you'll get what I've got above! And if you do it again, you should get:



See the pattern? If you're just starting this topic it might look strange, but do a few of these and this is what they all look like, more or less. After 5 years (2013), we have:



Pop that in your calculator, I get $9853.23!! For Part (b), you need to instead consider a general version of the expression, after \(n\) years:



We need \(A_n=0\) -> See if you can manipulate that expression to find \(n\)! If you've never done a question like this before let me know and I'd be happy to show you, or perhaps read this guide which steps through it for you! Happy to help if anything above was confusing as well - I'm assuming you've seen something similar to it before but can definitely go slower if you haven't :)

kauac

  • Forum Leader
  • ****
  • Posts: 554
  • Respect: +291
Re: Mathematics Question Thread
« Reply #3310 on: March 03, 2018, 08:23:25 pm »
0
How do you solve simultaneous equations when one of the equations is a cubic? Such as:
and
« Last Edit: March 03, 2018, 08:27:07 pm by kauac »
2018: HSC

2019: Gap Year

2020-2024: B Science / M Nutrition & Dietetics @ USYD

RuiAce

  • ATAR Notes Lecturer
  • Honorary Moderator
  • Great Wonder of ATAR Notes
  • *******
  • Posts: 8814
  • "All models are wrong, but some are useful."
  • Respect: +2575
Re: Mathematics Question Thread
« Reply #3311 on: March 03, 2018, 08:27:07 pm »
0
How do you solve simultaneous equations when one of the equations is a cubic? Such as:
and
You don't.

What's the full question?

kauac

  • Forum Leader
  • ****
  • Posts: 554
  • Respect: +291
Re: Mathematics Question Thread
« Reply #3312 on: March 03, 2018, 08:29:30 pm »
0
Find the area enclosed between y=x^3, x-axis and y= -3x+4?
2018: HSC

2019: Gap Year

2020-2024: B Science / M Nutrition & Dietetics @ USYD

RuiAce

  • ATAR Notes Lecturer
  • Honorary Moderator
  • Great Wonder of ATAR Notes
  • *******
  • Posts: 8814
  • "All models are wrong, but some are useful."
  • Respect: +2575
Re: Mathematics Question Thread
« Reply #3313 on: March 03, 2018, 08:34:53 pm »
0
Find the area enclosed between y=x^3, x-axis and y= -3x+4?
If that was it, then that is actually a very unfair question. You are NOT expected to solve general cubic equations in the HSC course, and it is unfair for them to make you guess that \(x=1\) is the solution (and hence \( (1,1) \) is the point of intersection.

kauac

  • Forum Leader
  • ****
  • Posts: 554
  • Respect: +291
Re: Mathematics Question Thread
« Reply #3314 on: March 03, 2018, 08:39:29 pm »
0
If that was it, then that is actually a very unfair question. You are NOT expected to solve general cubic equations in the HSC course, and it is unfair for them to make you guess that \(x=1\) is the solution (and hence \( (1,1) \) is the point of intersection.

Intriguing... there are a few similar questions with cubic equations in the textbook exercise that I am working on...Is there a certain way to approach these types of questions?
2018: HSC

2019: Gap Year

2020-2024: B Science / M Nutrition & Dietetics @ USYD