Science:Math Exam Resources/Courses/MATH101/April 2005/Question 06 (a)

MATH101 April 2005

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

   Other MATH101 Exams

  •  April 2011  •  April 2010  •  April 2009  •  April 2008  •  April 2007  •  April 2005  •  April 2006  •  April 2012  •  April 2013  •  April 2014  •  April 2015  •  April 2016  •  April 2018  •  April 2017  •  

Question 06 (a)

A tree trunk is 60 feet long and has circular cross sections. The diameters in feet measured at 10-foot intervals starting at the bottom of the tree are 10, 8, 7, 6, 5, 4, and 3. The volume of the tree, in cubic feet, is computed using the formula $V=\int _{0}^{60}A(y)\,dy,$ where $A(y)$ is the cross-sectional area of the trunk, in square feet, $y$ feet above the bottom of the tree. (a) Using a Simpson's Rule approximation for $V$, give an estimate for the volume of wood in the tree trunk. (Do not simplify your answer.)

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
To get the volume of the tree you will need to write a Simpson's Rule sum involving the cross sectional area of the trunk. Look closely at what you are given when evaluating the sum.

Hint 2
The formula for the Simpson's rule is given by $S_{n}:={\frac {\Delta x}{3}}\left(f(x_{0})+4f(x_{1})+2f(x_{2})+...+4f(x_{n-1})+f(x_{n})\right)$ where as always $n$ is an even integer.

Checking a solution serves two purposes: helping you if, after having used all the hints, 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

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. This is a straightforward application of Simpson's Rule to the integral $V=\int _{0}^{60}A(y)\,dy.$ Notice that ${\frac {\Delta x}{3}}={\frac {\tfrac {60-0}{6}}{3}}={\frac {10}{3}}$ By Simpson's Rule, the integral can be approximated by the sum ${\begin{aligned}V\approx {\frac {10}{3}}(A(0)+4A(10)+2A(20)+4A(30)+2A(40)+4A(50)+A(60))\end{aligned}}$ Remembering the given information about the diameter $D(y)$ of the tree trunk at various heights, $y$, we make the appropriate substitutions to determine the cross sectional area at each height: $A(y)=\pi \left({\frac {D(y)}{2}}\right)^{2}={\frac {\pi }{4}}D(y)^{2}$ Plugging into our sum we get the approximate volume for the tree trunk. ${\begin{aligned}V&\approx {\frac {10\pi }{12}}(10^{2}+4\times 8^{2}+2\times 7^{2}+4\times 6^{2}+2\times 5^{2}+4\times 4^{2}+3^{2})\end{aligned}}$

Click here for similar questions

MER QGH flag, MER QGQ flag, MER QGS flag, MER QGT flag, MER Tag Simpson's rule, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag