Science:Math Exam Resources/Courses/MATH102/December 2015/Question 12

MATH102 December 2015

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Question 12
The function $f(x)=3{\sqrt {x}}+20$ models the median height (in inches) of boys who are $x$ months of age. Suppose Lucas’s growth follows this pattern. Use linear approximation to predict Lucas’ height on his 2nd birthday (i.e. age 24 months).

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
Use the linear approximation of $f$ at $a=25$.

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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. To find $f(24)$, we use the linear approximation of $f$ at $a=25$. (because its square root is simple; ${\sqrt {25}}=5$) For this purpose, we find its derivative and evaluate at $x=25$; $f'(x)=(3{\sqrt {x}}+20)'={\frac {3}{2{\sqrt {x}}}}$ and hence $f'(25)={\frac {3}{2{\sqrt {25}}}}={\frac {3}{10}}$. Then, the linear approximation is $f(x)\sim f'(25)(x-25)+f(25)={\frac {3}{10}}(x-25)+(3{\sqrt {25}}+20)={\frac {3}{10}}(x-25)+35$ Therefore $f(24)\sim {\frac {3}{10}}(24-25)+35=-{\frac {3}{10}}+35={\frac {347}{10}}=34.7$ Answer: $\color {blue}34.7inches$