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[SOLVED] modeling the removal of waste from a national park

DeusAbscondus

Active member
Jun 30, 2012
176
Hi folks,
This problem is from a trial test (the main test is in one week's time)
Quickly for some background to what I have done maths-wise:
1. a crash course in college level maths in the first semester of Australian academic year: (february 2012 to June 2012) followed by crash course in calculus (july till now) This, just to give you an idea of the level I'm at.

Here is the question/problem, verbatim:

"Data has been recorded over the past 10 years measuring the quantity of litter, Q, removed from a particular park. Over the period $$\frac{dQ}{dt} \text{ is less than 0 and} \frac{d^2Q}{dt^2}\text{ is greater than 0 }$$

i) Draw a neat sketch of Q against t over the last 10 years.
ii) What conclusions can be drawn about the amount of litter over this period?

I deduce from this that:
1. $f(x)$ must slope negatively for the 10 year period (placing the tangent curve below the x-axis) as f'(x) has been less than zero for that period; and
2. the concavity has been positive for the entire period (f''(x) is greater than zero) meaning that f(x) is bending concave up the whole of this time.

Finally, to graph f(x), I chose an abitrary point (what choice did I have here?) up the Q axis, and sketched part of a concave up parabola (a curve whose slope is decreasing, decreasingly) curving down towards Q=0 at point t=10.

Question 1: am i justified in my first 2 assumptions?
Question 2: would my graph be a fair representation of what we know from the problem?
Question 3: is there any reason to support the hypothesis that f(x) is quadratic, f'(x) linear negative and f''(x) positive constant?

thanks guys,
Deus Abs
PS Latex for "less than" and "greater than" ?
 

CaptainBlack

Well-known member
Jan 26, 2012
890
Hi folks,
This problem is from a trial test (the main test is in one week's time)
Quickly for some background to what I have done maths-wise:
1. a crash course in college level maths in the first semester of Australian academic year: (february 2012 to June 2012) followed by crash course in calculus (july till now) This, just to give you an idea of the level I'm at.

Here is the question/problem, verbatim:

"Data has been recorded over the past 10 years measuring the quantity of litter, Q, removed from a particular park. Over the period $$\frac{dQ}{dt} \text{ is less than 0 and} \frac{d^2Q}{dt^2}\text{ is greater than 0 }$$

i) Draw a neat sketch of Q against t over the last 10 years.
ii) What conclusions can be drawn about the amount of litter over this period?

I deduce from this that:
1. $f(x)$ must slope negatively for the 10 year period (placing the tangent curve below the x-axis) as f'(x) has been less than zero for that period; and
2. the concavity has been positive for the entire period (f''(x) is greater than zero) meaning that f(x) is bending concave up the whole of this time.

Finally, to graph f(x), I chose an abitrary point (what choice did I have here?) up the Q axis, and sketched part of a concave up parabola (a curve whose slope is decreasing, decreasingly) curving down towards Q=0 at point t=10.

Question 1: am i justified in my first 2 assumptions?
Question 2: would my graph be a fair representation of what we know from the problem?
Question 3: is there any reason to support the hypothesis that f(x) is quadratic, f'(x) linear negative and f''(x) positive constant?

thanks guys,
Deus Abs
PS Latex for "less than" and "greater than" ?
There is no reason to assume that \(Q=0\) at \(t=10\), that would imply no litter at the current time. Also there is no reason to suppose that Q is even heading towards zero it may level out at some positive value.

1. Yes
2. There is no reason to assume that the curve will ever reach the x-axis, or even approach it.
3. There is no reason to suppose the curve is quadratic, in fact there are good reasons to suppose otherwise. Probably something of the form: \(Q(t)=(Q(0)-Q(\infty))\exp(-\lambda t)+Q(\infty)\) with \(Q(0)>Q(\infty)\) would be better.

CB
 

DeusAbscondus

Active member
Jun 30, 2012
176
There is no reason to assume that \(Q=0\) at \(t=10\), that would imply no litter at the current time. Also there is no reason to suppose that Q is even heading towards zero it may level out at some positive value.

1. Yes
2. There is no reason to assume that the curve will ever reach the x-axis, or even approach it.
3. There is no reason to suppose the curve is quadratic, in fact there are good reasons to suppose otherwise. Probably something of the form: \(Q(t)=(Q(0)-Q(\infty))\exp(-\lambda t)+Q(\infty)\) with \(Q(0)>Q(\infty)\) would be better.

CB
Thanks Cap'n. The answers above confirm that I am on the right track and correct some false assumpitons.
However, this is lost on me as I have no idea what it means:
$$Q(t)=(Q(0)-Q(\infty))\exp(-\lambda t)+Q(\infty) \text{ with }Q(0)>Q(\infty)$$

If you were to explicate this a bit I would be appreciative.

(I presume that the expression assumes knowledge of the integral, of which, in the famous words of Sergeant Schultz: "I know nothing")

Deus Abs
 

CaptainBlack

Well-known member
Jan 26, 2012
890
Thanks Cap'n. The answers above confirm that I am on the right track and correct some false assumpitons.
However, this is lost on me as I have no idea what it means:
$$Q(t)=(Q(0)-Q(\infty))\exp(-\lambda t)+Q(\infty) \text{ with }Q(0)>Q(\infty)$$

If you were to explicate this a bit I would be appreciative.

(I presume that the expression assumes knowledge of the integral, of which, in the famous words of Sergeant Schultz: "I know nothing")

Deus Abs
It means that at \(t=0\) the quantity of litter removed over an interval is \(Q(0)\) and that as \(t \to \infty; \ Q(t) \to Q(\infty)\) and the proposed form has the shape specified.

(Put \(Q(0)=0.5\), \(Q(\infty)=1/10\) and \(\lambda=0.1\) and plot the function to see what is going on.


Click on the graph below to see in more detail:
[graph]ldvtqukuuc[/graph]

CB
 
Last edited:

DeusAbscondus

Active member
Jun 30, 2012
176
It means that at \(t=0\) the quantity of litter removed over an interval is \(Q(0)\) and that as \(t \to \infty; \ Q(t) \to Q(\infty)\) and the proposed form has the shape specified.

(Put \(Q(0)=0.5\), \(Q(\infty)=1/10\) and \(\lambda=0.1\) and plot the function to see what is going on.


Click on the graph below to see in more detail:
[graph]ldvtqukuuc[/graph]

CB
Thanks kindly Cap'n.
Not least for introducing me to the fantastic graphics calculator at Desmos!

Regs,
Deus Abs