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

MATH110 April 2012

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Question 06 (a)

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 $\displaystyle {\frac {dP}{dt}}=10^{8}P(t).$ (a) Write down an exponential algebraic expression for P(t) which satisfies the differential equation given above.

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
Which function do you know whose derivative is the same as the original function?

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The differential equation above says that the derivative of the function $P(t)$ is a constant multiple of the original function. A function that has this property is the exponential function, $e^{t}$. An exponential function will satisfy the differential equation when its derivative is equal to $10^{8}$ times the same exponential function. This occurs when we include a factor of $10^{8}$ in the exponent of the $e^{t}$ so we have the function $P(t)=e^{10^{8}t}$. We can check by differentiation that this satisfies the differential equation ${\frac {dP}{dt}}=10^{8}P$.

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