Mathematics Question Thread

[RuiAce]

Hi, I'm struggling to find a value for t in which ln 0.2t = 5t.
Any help would be much appreciated.

It is essentially as Clovvy has stated. In general, there is no algebraic method whatsoever of solving equations of this form. (In fact, \(bx = f(x)\) can not be solved algebraically at all, where \(f(x)\) is any exponential, logarithmic or trigonometric function.)

Therefore, I am not sure where you got this question from.

Note that there is also some difficulty using the graphical method for these types of problems. This is because \( ax = \ln bx\) will have solutions for some values of \(a\) and \(b\), but not for all possible values. Therefore, it is very hard to tell just by staring at the equation of the graph, whether or not there will be points of intersection between the two.

[kauac]

Hi...
More of an exam technique question:

Is it generally better to try and do the paper as fast as possible, to maximize time spent checking over, or do it a bit slower and check as you go?

When I do past papers with the first approach, I generally have about 20-30min check-over time, but then I don't always pick up on the silly errors I make. But I'm worried if I try and do it slower and check as I go, that I will run out of time.

[RuiAce]

Hi...
More of an exam technique question:

Is it generally better to try and do the paper as fast as possible, to maximize time spent checking over, or do it a bit slower and check as you go?

When I do past papers with the first approach, I generally have about 20-30min check-over time, but then I don't always pick up on the silly errors I make. But I'm worried if I try and do it slower and check as I go, that I will run out of time.

The most optimal balance varies from person to person. It requires balancing out factors including (but not necessarily limited to) how error prone you are, and your time management skills in the exam.

For me, I try to avoid checking immediately after I do a question and taking it too slow. More or less because I know I'll have preconceptions about what I think is right, and then not spot it so easily in comparison to when I check it later. But I do choose to check the paper the instant I hit that point where all the remaining marks are no longer so obvious to me. That way, at least I'm checking the paper with reasonable amount of time left, but also at a point where I'm comfortable (for now) with the amount of marks I had already scored.

[Sr-1425]

Oops, my bad. I miscalculated and got stuck.
I got ln (0.2/t) = -1.1t, but I'm not sure how to continue from there.

[RuiAce]

Oops, my bad. I miscalculated and got stuck.
I got ln (0.2/t) = -1.1t, but I'm not sure how to continue from there.

Still cannot be solved algebraically. As stated, no equation with logs on one side and linear terms on the other can be solved algebraically.

Please provide the original question in its entirety.

[Mate2425]

Hey guys for rates questions, when they ask what is the 'greatest' e.g. in HSC 2015 Q15c , is it always when it hits the x-intercept as for this question it seemed to be the case but in HSC 2016 Q16b they took the halfway distance between the two roots (= maximum peak of a parabola).
Is there a standardised process to 'Find the greatest' questions?

Thank you 

[RuiAce]

Hey guys for rates questions, when they ask what is the 'greatest' e.g. in HSC 2015 Q15c , is it always when it hits the x-intercept as for this question it seemed to be the case but in HSC 2016 Q16b they took the halfway distance between the two roots (= maximum peak of a parabola).
Is there a standardised process to 'Find the greatest' questions?

Thank you 

They are asking for slightly different things.

In the 2015 paper, they want you to maximise the quantity, i.e. the volume. We know from doing classical maxima/minima problems that \(V\) can be maximised by setting \( \frac{dV}{dt} = 0\), and solving for the time \(t\). This is the usual method.

But in the 2016 paper, they don't want you to maximise the quantity, i.e. the population itself. They want you to maximise the rate of growth of the quantity, i.e. the rate of growth of the population. To maximise \( \frac{dy}{dt} \), in theory what we would want is to set \( \frac{d^2y}{dt^2} = 0\). But because we don't know how to obtain that (because we can't differentiate with respect to \(t\) here anymore), we can just try maximising \( \frac{dy}{dt} \) by literally plotting the graph of \( \frac{dy}{dt} \) v.s. \(y\). From there, we can read the value of \(y\) (NOT \(t\)) that maximises \( \frac{dy}{dt} \).

[Mate2425]

They are asking for slightly different things.

In the 2015 paper, they want you to maximise the quantity, i.e. the volume. We know from doing classical maxima/minima problems that \(V\) can be maximised by setting \( \frac{dV}{dt} = 0\), and solving for the time \(t\). This is the usual method.

But in the 2016 paper, they don't want you to maximise the quantity, i.e. the population itself. They want you to maximise the rate of growth of the quantity, i.e. the rate of growth of the population. To maximise \( \frac{dy}{dt} \), in theory what we would want is to set \( \frac{d^2y}{dt^2} = 0\). But because we don't know how to obtain that (because we can't differentiate with respect to \(t\) here anymore), we can just try maximising \( \frac{dy}{dt} \) by literally plotting the graph of \( \frac{dy}{dt} \) v.s. \(y\). From there, we can read the value of \(y\) (NOT \(t\)) that maximises \( \frac{dy}{dt} \).

Thanks Rui, 

[Sr-1425]

This is the question:

The concentration of a certain drug in the blood at a time t hours after taking a dose is x units, where x=0.3t*(e)^(-1.1t)
A) Determine the maximum concentration and the time at which this is reached.
B) Plot the function for t=0, 0.1, 0.5, 1, 2, 3.
C) This drug kills germs only while its concentration is at least 0.06 units. From the graph, find the length of time during which the drug will kill germs.


It's part c) where I can't seem to find a solution

[RuiAce]

This is the question:

The concentration of a certain drug in the blood at a time t hours after taking a dose is x units, where x=0.3t*(e)^(-1.1t)
A) Determine the maximum concentration and the time at which this is reached.
B) Plot the function for t=0, 0.1, 0.5, 1, 2, 3.
C) This drug kills germs only while its concentration is at least 0.06 units. From the graph, find the length of time during which the drug will kill germs.


It's part c) where I can't seem to find a solution

Because it says "from the graph", you aren't being asked to determine the value of \(t\) algebraically. You're being asked to read off your graph and interpret an approximate value.

In fact, given how informal part b) is, only asking you to plot some dots, it's really up to you to infer the shape of the graph itself. In the HSC you would not be asked a question as vague as this - they will run you through the usual steps of stationary points if you had to do a plot.

But otherwise, you can draw the curve however you want to draw it. From Desmos output, I approximate that \(t \approx 0.3\) and \(t \approx 2.2\), so that would be a length of 1.9 hours.

[Sr-1425]

Because it says "from the graph", you aren't being asked to determine the value of \(t\) algebraically. You're being asked to read off your graph and interpret an approximate value.

In fact, given how informal part b) is, only asking you to plot some dots, it's really up to you to infer the shape of the graph itself. In the HSC you would not be asked a question as vague as this - they will run you through the usual steps of stationary points if you had to do a plot.

But otherwise, you can draw the curve however you want to draw it. From Desmos output, I approximate that \(t \approx 0.3\) and \(t \approx 2.2\), so that would be a length of 1.9 hours.

(Image removed from quote.)

Thank you.

[LaraC]

Hello,

I have a really simple question that I can't work out.....its only at the start of the paper is there just a rule I'm forgetting?

Solve the equation lnx=2. Give your answer correct to four decimal places.

Thanks

[RuiAce]

Hello,

I have a really simple question that I can't work out.....its only at the start of the paper is there just a rule I'm forgetting?

Solve the equation lnx=2. Give your answer correct to four decimal places.

Thanks

\[ \ln x = 2\text{ becomes }\boxed{x = e^2}\\ \text{so just plug }e^2\text{ in your calculator to obtain }7.3891\]
(rounded up)

[LaraC]

But how do u just 'know' that x = e^2? 
Sorry....i should know

[fun_jirachi]

Log laws
When

it's the same as
.
Just put in the numbers and you get your x=e²