July 13, 2020, 07:14:56 pm

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#### RuiAce

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« Reply #4230 on: June 03, 2019, 06:30:47 pm »
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Thank you for the tree diagram! Will the HSC ask us to draw a complex one like that? Because it takes quite some time to get the answers.
Will comment that I've never seen a tree diagram be demanded ever. It's purely there as a reference tool (and can grab you marks for working out when done appropriately).

It's heavily advised at the start (because otherwise people get lost too easily), but every probability question can ultimately be done without a tree diagram. But as a general rule of thumb, the more complicated the questions get, the nastier tree diagrams become to draw. Whilst there are some ways to overcome it:
- at some point you just get the idea and cannot be bothered to draw everything else, so you only draw like half of the tree diagram (or some incomplete tree diagram)
- be more selective of what your labels are, constantly adjusting them as appropriate, rather than relying on one generic tree diagram
at the end of the day, they are not necessary for any question regardless of context.

If your teacher enforced the tree diagram for internal assessment, I would say that it's for your own benefit since they're giving you good habits and you should do it. I just would not happen to agree with that teaching philosophy because I know many bright students can just do without. For the final exam, this won't be something you need to worry about.

#### therese07

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« Reply #4231 on: June 03, 2019, 09:04:23 pm »
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Hi there!

I'm doing some revision for integration trig functions, and I'm really struggling to do this question!

All help is appreciated!
2020: Bachelor of Arts/Bachelor of Law @ Macquarie University

#### fun_jirachi

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« Reply #4232 on: June 03, 2019, 11:22:54 pm »
+2
Hey there!

Remember that the volume when rotated around the x-axis is defined by
$V=\pi\int_a^b [f(x)]^2 \ dx \\ \text{In this case, it is} \ \pi\int_0^{0.15} \sec^2{\pi x} \ dx \\ =\pi \times \left[\frac{\tan\pi x}{\pi}\right]_0^{0.15}$

You should be able to take it from here
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#### Thankunext

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« Reply #4233 on: June 04, 2019, 10:51:54 am »
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Hello! I need help with the interpretation of this question as I believe this is without replacement, however the answers suggest that it is with replacement.
The ratio of girls to boys at a school is four to five. Two students are surveyed at random from the school. Find the probability that the students are
A) both boys
B) a girl and a boy
C) at least one girl.
A) 25/81
B) 40/81
C) 56/81
I had the same problem with most of the questions from this exercise as by common sense I think it is without replacement but maybe I am reading the question wrong??

#### RuiAce

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« Reply #4234 on: June 04, 2019, 11:31:12 am »
+2
Hello! I need help with the interpretation of this question as I believe this is without replacement, however the answers suggest that it is with replacement.
The ratio of girls to boys at a school is four to five. Two students are surveyed at random from the school. Find the probability that the students are
A) both boys
B) a girl and a boy
C) at least one girl.
A) 25/81
B) 40/81
C) 56/81
I had the same problem with most of the questions from this exercise as by common sense I think it is without replacement but maybe I am reading the question wrong??
These questions have an extra subtlety behind them. CSSA papers love doing this but they try to make it a bit clearer.
$\text{The hidden assumption is that we're considering}\\ \text{a }\textbf{very large}\text{ sample.}\\ \text{Just for the sake of example, let's assume that the school}\\ \text{has 900 students.}$
$\text{Then there would be 400 girls and 500 boys.}\\ \text{Suppose we want the probability of them both being boys.}\\ \text{Then the answer would be }\frac{500}{900}\times \frac{499}{899}.$
In a similar way, let's say it has 1800 students (an arguably giant school). Then the probability would be $\frac{1000}{1800} \times \frac{999}{1799}$.
$\text{Observe how technically speaking, the probability actually changes}\\ \text{a little when we change the number of students at the school.}$
$\text{However if you plug into the calculator, you'll see that }\frac{499}{899} \approx \frac{500}{900} = 5/9\text{ anyway.}\\ \text{Similarly, }\frac{999}{1799}\approx \frac{1000}{1800} = \frac59.$
Here's where everyone gets confused. People tend to think that because the ratio is $4:5$, there should only be $9$ students. Thus they're lead to think that the required probability should be $\frac{5}{9} \times \frac{4}{8}$.

When written in English, this makes no sense at all. But it's easy to miss this subtlety in the question because a lot of people forget that a ratio tells us nothing about how big the original sample is!
$\text{This is why we usually assume that the sample is very large.}\\ \text{When we do so, we see that after surveying one student}\\ \text{the ratio of remaining students isn't necessarily equal to }4:5\text{ anymore},\\ \text{but is still }\textbf{approximately}\text{ equal to }4:5.$
$\text{So we }\textbf{estimate}\text{ our required probability under this assumption.}\\ \text{This is why we arrive at }\frac59 \times \frac59\text{ instead.}$
Note that these probabilities can never be computed to be exact. But at some point the error in the approximation is just negligible so we assume it doesn't matter. This question wasn't the best in my opinion because I feel uncomfortable assuming schools have 10000+ people, but it still illustrate the idea.
« Last Edit: June 04, 2019, 12:06:09 pm by jamonwindeyer »

#### Kombmail

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« Reply #4235 on: June 04, 2019, 08:19:38 pm »
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I was wondering what I need to prepare in terms of using exponentials and logs in applications of calculus to the physical world and series and sequences since i have a test on the 21st on these things!

What should I note rather than revising exponentials and log seperately since I need time to do past papers since that it what I will be tested upon in my exam.

Any suggestions?
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#### Georgakopoulou

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« Reply #4236 on: June 04, 2019, 08:57:52 pm »
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Hey! Attached I have a question I came across and had a bit of trouble with. It would be greatly appreciated if someone could help me!

« Last Edit: June 04, 2019, 09:19:11 pm by Georgakopoulou »

#### MB_

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« Reply #4237 on: June 05, 2019, 10:17:17 am »
+3
Hey! Attached I have a question I came across and had a bit of trouble with. It would be greatly appreciated if someone could help me!
\begin{align*}\text{The revenue is } R(t) &= 1000000 \cdot 0.5 \cdot (1-e^{-0.04t})\\ &= 500000(1-e^{-0.04t})\\ \text{The cost is } C(t) &= 1000t\\~\\ \therefore \text{ The profit is } P(t) &= 500000(1-e^{-0.04t})-1000t \end{align*}
Now you have to find $t$ that maximizes profit by differentiating the profit function, setting it equal to $0$ and solving for $t$.
« Last Edit: June 05, 2019, 06:09:05 pm by MB_ »
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#### Georgakopoulou

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« Reply #4238 on: June 05, 2019, 02:17:03 pm »
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Thank you so much!

\begin{align*}\text{The revenue is } R(t) &= 1000000 \cdot 0.5 \cdot (1-e^{-0.04t})\\
&= 500000(1-e^{-0.04t})\\
\text{The cost is } C(t) &= 1000t\\~\\
\therefore \text{ The profit is } P(t) &= 500000(1-e^{-0.04t})-1000t
\end{align*}
Now you have to find $t$ that maximizes profit by differentiating the profit function, setting it equal to $0$ and solving for $t$.

#### mirakhiralla

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« Reply #4239 on: June 11, 2019, 10:22:08 pm »
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When a q says "using one application of Simpsons rule, find the area", how many function values is that?

#### RuiAce

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« Reply #4240 on: June 11, 2019, 10:24:18 pm »
+2
When a q says "using one application of Simpsons rule, find the area", how many function values is that?
3.

If $N$ is the number of applications of Simpson's rule, you'll be using $2N+1$ function values.

#### mirakhiralla

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« Reply #4241 on: June 11, 2019, 10:25:26 pm »
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3.

If $N$ is the number of applications of Simpson's rule, you'll be using $2N+1$ function values.

is that the same for trapezoidal?

#### RuiAce

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« Reply #4242 on: June 11, 2019, 10:28:00 pm »
+3

is that the same for trapezoidal?
I haven't seen that wording be used for trapezoidal rule before. It's quite ambiguous, because one application of the trapezoidal rule usually means only one trapezium (i.e. one sub-interval). Yet some may argue it actually requires two sub-intervals as well (i.e. 3 function values) because the formula is written as such.

Usually, for the trapezoidal rule, the number of sub-intervals is specified instead to make it abundantly clear what is required.

#### mirakhiralla

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« Reply #4243 on: June 11, 2019, 10:31:32 pm »
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I haven't seen that wording be used for trapezoidal rule before. It's quite ambiguous, because one application of the trapezoidal rule usually means only one trapezium (i.e. one sub-interval). Yet some may argue it actually requires two sub-intervals as well (i.e. 3 function values) because the formula is written as such.

Usually, for the trapezoidal rule, the number of sub-intervals is specified instead to make it abundantly clear what is required.

okay thank you so much

#### Kombmail

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