# Sketching a graph of contaminants after a train crash

Bhawna L
Homework Statement:
After a train derailment in Northern Ontario, the concentration, C, in grams per litre, of a pollutant after t minutes in a 5 000 000-L pond can be modelled by the function C(t) =30t/200000 +t, when a pollutant concentration of 30 g/L flows into the pond at a rate of 25 L/min. a) Sketch a graph showing the concentration of the pollutant after t minutes
Relevant Equations:
See above
I tried to graph this equation, but all I am getting is a straight line that is x=0. I don't understand what I am doing wrong.

Staff Emeritus
Gold Member
How are you trying to graph it? Like on a calculator or something?

Bhawna L
How are you trying to graph it? Like on a calculator or something?
Yeah, I tried using an online graphing calculator, but all I'm getting is x=0, which is not what is supposed to be based on the answers that I was given. I am just trying to figure out how to actually graph it.

Gold Member
I'm kind of confused by your question.
Are you asking how to graph the function ##C(t)=\frac{30⋅t}{200000}+t=(\frac{200030}{200000})⋅t##?

Bhawna L
I'm kind of confused by your question.
Are you asking how to graph the function ##C(t)=\frac{30⋅t}{200000}+t=(\frac{200030}{200000})⋅t##?
The function is ##C(t)=\frac{30⋅t}{200000+t}##

Gold Member
The function is ##C(t)=\frac{30⋅t}{200000+t}##
OK, got it. Mind your parentheses, that's not the function you originally typed.

If you are asking how to work a graphing calculator, I am neither interested or capable of helping. But I can help a bit in understanding what this function looks like, with some tricks about graphing in general.

1) Identify the domain of the function, i.e. what values of ##t## do you care about. In this case I'll guess that it's ##0 \leq t \leq \infty##. Then evaluate the function at those limits and put those points on your graph. For infinity, you can put in a really big value for ##t##. Even though you can't graph it at ##\infty##, do it and write down the value.

2) Identify factors in the function that are simple operations on a more basic function. For example ##y=x^2+2##, or ##y=10x^2##, are really just simple variations on ##y=x^2##. Don't waste too much effort on these numbers, just use as required. So from an abstract point of view your function is similar to ##y=\frac{t}{t+a}##, with ##a=200000##, just 30 times bigger.

3) Try to see how the function behaves near the extreme values, you can often approximate the function with a simpler version. For example, in this case, what is the difference between ##C(t)=\frac{30⋅t}{200000+t}## and ##C(t)=\frac{30⋅t}{200000}## when ##t \ll 200000##, for example ##t=1##, ##t=10##, ##t=100##, etc. Same for the large values (##t \gg 200000##, like ##t=10^8##), but a different approximation. This will give you two asymptotic approximate functions that will be very close to the real function at the extremes, but won't work so well in the middle.

4) Sketch those asymptotic functions (dotted lines, usually) as guidelines for where/how the function starts and ends.

5) In the middle section, where ##t \ll 200000## and ##t \gg 200000## aren't true, you'll need to calculate some values and plot them to see how the function makes the transition from one asymptote to the other. I'd start with ##t=200000##, as a good guess.

OK, now the disclaimer. This process doesn't always work. Some functions are a pain in the @##. But you'll probably recognize most of those, or figure out in this process that things aren't simple. For well behaved functions, you can just draw a smooth line to connect the points you calculated and approaching the asymptotes.

Finally, as an enticement: this process will be easier for you in the future when you've learned about calculus and other math stuff.

docnet and Bhawna L
Bhawna L
OK, got it. Mind your parentheses, that's not the function you originally typed.

If you are asking how to work a graphing calculator, I am neither interested or capable of helping. But I can help a bit in understanding what this function looks like, with some tricks about graphing in general.

1) Identify the domain of the function, i.e. what values of ##t## do you care about. In this case I'll guess that it's ##0 \leq t \leq \infty##. Then evaluate the function at those limits and put those points on your graph. For infinity, you can put in a really big value for ##t##. Even though you can't graph it at ##\infty##, do it and write down the value.

2) Identify factors in the function that are simple operations on a more basic function. For example ##y=x^2+2##, or ##y=10x^2##, are really just simple variations on ##y=x^2##. Don't waste too much effort on these numbers, just use as required. So from an abstract point of view your function is similar to ##y=\frac{t}{t+a}##, with ##a=200000##, just 30 times bigger.

3) Try to see how the function behaves near the extreme values, you can often approximate the function with a simpler version. For example, in this case, what is the difference between ##C(t)=\frac{30⋅t}{200000+t}## and ##C(t)=\frac{30⋅t}{200000}## when ##t \ll 200000##, for example ##t=1##, ##t=10##, ##t=100##, etc. Same for the large values (##t \gg 200000##, like ##t=10^8##), but a different approximation. This will give you two asymptotic approximate functions that will be very close to the real function at the extremes, but won't work so well in the middle.

4) Sketch those asymptotic functions (dotted lines, usually) as guidelines for where/how the function starts and ends.

5) In the middle section, where ##t \ll 200000## and ##t \gg 200000## aren't true, you'll need to calculate some values and plot them to see how the function makes the transition from one asymptote to the other. I'd start with ##t=200000##, as a good guess.

OK, now the disclaimer. This process doesn't always work. Some functions are a pain in the @##. But you'll probably recognize most of those, or figure out in this process that things aren't simple. For well behaved functions, you can just draw a smooth line to connect the points you calculated and approaching the asymptotes.

Finally, as an enticement: this process will be easier for you in the future when you've learned about calculus and other math stuff.
Thank you! Really appreciate it.

Homework Helper
Gold Member
You can copy-paste your equation into https://www.desmos.com/calculator
rather quickly and see what is going on. Use parentheses as needed.
You may have to scale the axes (use the wrench to set values or
shift-mouseDrag on the axes or along the edge for axes out of the viewport).

It might be helpful to clarify the units being used in the equation
to see that the parentheses are in the correct places.

Mentor
Homework Statement:: After a train derailment in Northern Ontario, the concentration, C, in grams per litre, of a pollutant after t minutes in a 5 000 000-L pond can be modeled by the function C(t) =30t/200000 +t,
Because of missing parentheses, your equation was interpreted to mean this:
$$C(t) = \frac{30t}{200000} + t$$
The function is ##C(t)=\frac{30⋅t}{200000+t}##
Much better.
Mind your parentheses, that's not the function you originally typed.

It looks like you (@Bhawna L) have figured out how to use LaTeX to write this equation correctly. If you write fractions inline, it's important to use parentheses in the right places. For your equation, it would be C(t) = 30t/(200,000 + t).