Science:Math Exam Resources/Courses/MATH110/April 2012/Question 06 (b)

{{#incat:MER QGQ flag|{{#incat:MER QGH flag|{{#incat:MER QGS flag|}}}}}}

MATH110 April 2012

  •  Q1  •  Q2 (a)  •  Q2 (b)  •  Q3 (a)  •  Q3 (b)  •  Q4  •  Q5 (a)  •  Q5 (b)  •  Q5 (c)  •  Q5 (d)  •  Q5 (e)  •  Q5 (f)  •  Q5 (g)  •  Q6 (a)  •  Q6 (b)  •  Q7  •  Q8  •  Q9  •  

   Other MATH110 Exams

  •  December 2011  •  December 2010  •  April 2012  •  April 2011  •  December 2012  •  April 2013  •  December 2013  •  April 2014  •  December 2014  •  December 2015  •  April 2017  •  April 2016  •  December 2016  •  December 2017  •  April 2018  •  

Question 06 (b)

In a fission nuclear weapon, a sphere of uranium-235 is compressed to a fraction of its size, forcing at least one atomic nucleus to split and release neutrons. These neutrons force other nuclei to release their neutrons, and so on. The result is that the number of neutrons grows exponentially. Let P(t) be the number of neutrons t seconds after detonation. The number of neutrons satisfies the differential equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle \displaystyle \frac{dP}{dt} = 10^8P(t). } (b) According to your model in part (a), how long does it take a given number of neutrons to double?

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Say you started with 5 neutrons (or 500, or 5 million) - how would you incorporate this information into your formula from part (a)?

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

We will call our given number of neutrons at the start (when t = 0), Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle P_0} . Note that when we calculate the number of neutrons at the start, we get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle P(0) = 1} . So in order for our function Failed to parse (Conversion error. Server ("https://wiki.ubc.ca/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle P(t)} to give us the starting number of neutrons, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle P_0} , we need to multiply Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle e^{10^8t}} by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle P_0} to get our final formula, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle P(t) = P_0e^{10^8t}} . To calculate how long it takes for the neutrons to double, we set our function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle P(t)} equal to double the starting amount, or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle 2P_{0}} . This gives us Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle 2P_0 = P_0e^{10^8t}} After canceling the Failed to parse (Conversion error. Server ("https://wiki.ubc.ca/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle P_{0}} from both sides, taking a natural logarithm of both sides, and simplifying using log rules we get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle \ln(2) = \ln(e^{10^8t})} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle \displaystyle \ln(2)=10^{8}t} Failed to parse (Conversion error. Server ("https://wiki.ubc.ca/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {\ln(2)}{10^{8}}}=t}

{{#incat:MER CT flag||

Click here for similar questions

MER QGH flag, MER QGQ flag, MER QGS flag, MER QGT flag, MER Tag Population dynamics, MER Tag Separation of variables, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag

}}