Science:Math Exam Resources/Courses/MATH103/April 2009/Question 08 (a)

MATH103 April 2009

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Question 08 (a)
The growth rate of a plant during the first week after germination is given by $\ g(t)=ae^{t/2}{\text{ mm/day}},$ where the time t is measured in days, and a is a constant. Determine the height of the plant h(t) as a function of time t, assuming that it was zero at time t = 0.

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
How are the height and the growth rate related?

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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. Since the growth rate is the derivative of the height for any time $\ t\geq 0$, we need to integrate the function $\ g(t)$ to find the height depending on $\ t$. $h(t)=\int g(t)dt=\int ae^{\frac {t}{2}}dt=2ae^{\frac {t}{2}}+c$ for a constant $\ c\in \mathbb {R}$. We find that constant by using the initial condition. $0=h(0)=2a+c,\$ $c=-2a\$. So, the function of height is $h(t)=2a(e^{\frac {t}{2}}-1)\$.

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