Sketching a graph of contaminants after a train crash

In summary: I'd start with ##t=200000##, as a good guess.In summary, 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.
  • #1
Bhawna L
4
0
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.
 
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  • #2
How are you trying to graph it? Like on a calculator or something?
 
  • #3
Office_Shredder said:
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.
 
  • #4
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##?
 
  • #5
DaveE said:
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}##
 
  • #6
Bhawna L said:
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.
 
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Likes docnet and Bhawna L
  • #7
DaveE said:
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.
 
  • #8
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.
 
  • #9
Bhawna L said:
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:
$$C(t) = \frac{30t}{200000} + t$$
Bhawna L said:
The function is ##C(t)=\frac{30⋅t}{200000+t}##
Much better.
DaveE said:
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).
 

1. How do you determine the contaminants present in a train crash?

The contaminants present in a train crash can be determined through various methods such as conducting a chemical analysis of the debris and soil samples, using remote sensing techniques, and monitoring air and water quality in the affected area.

2. What factors affect the spread of contaminants after a train crash?

The spread of contaminants after a train crash can be affected by several factors including the type and quantity of hazardous materials involved, weather conditions, topography of the area, and the response and cleanup efforts by emergency personnel.

3. How do you create a graph of contaminants after a train crash?

To create a graph of contaminants after a train crash, you will need to collect data on the concentration of contaminants at various locations and time intervals. This data can then be plotted on a graph using appropriate software or by hand. The graph should have the concentration of contaminants on the y-axis and time or distance on the x-axis.

4. What is the purpose of sketching a graph of contaminants after a train crash?

The purpose of sketching a graph of contaminants after a train crash is to visually represent the spread and concentration of hazardous materials in the affected area. This can help in identifying hotspots of contamination, monitoring the effectiveness of cleanup efforts, and informing the public and authorities about potential health risks.

5. How can the information from a graph of contaminants after a train crash be used?

The information from a graph of contaminants after a train crash can be used by emergency responders to prioritize and plan cleanup efforts, by environmental agencies to assess the extent of contamination and its impact on the environment, and by the public to understand potential health risks and take necessary precautions.

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