Error Propagation Area Of Triangle
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Error Propagation Calculus
voted up and rise to the top Area of triangle and uncertainty estimation up vote 2 down vote favorite Heron's formula states that if a plane triangle has sides $a,b{\text{ and }}c$, then its area is given by $A = \sqrt {s(s - a)(s - b)(s - c)} $, where $s = \frac{1}{2} \cdot (a + b + c)$ is half the circumference of the triangle. This can also be error propagation khan academy expressed as $$A(a,b,c) = \frac{1}{4} \cdot \sqrt {2({a^2}{b^2} + {b^2}{c^2} + {c^2}{a^2}) - ({a^4} + {b^4} + {c^4})}. $$ The sides of a triangle were measured to be $10.0 \pm 0.1{\text{ m}}{\text{,}}\,\,\,17.0 \pm 0.3{\text{ m}}{\text{,}}\,\,\,{\text{21}}{\text{.0}} \pm {\text{0}}{\text{.4 m}}$. Use differentials to calculate an approximate upper limit for the uncertainty in the approximation $A \approx 84.0{\text{ }}{{\text{m}}^2}$ due to the uncertainties in the measurements of the side lengths $a,b$ and $c$. My attempt: $$\begin{gathered} A = \frac{1}{4}\sqrt {2({a^2}{b^2} + {b^2}{c^2} + {c^2}{a^2}) - ({a^4} + {b^4} + {c^4})} ;\,\,\,\,\,\partial A = \frac{{\partial A}}{{\partial a}} \cdot \partial a + \frac{{\partial A}}{{\partial b}} \cdot \partial b + \frac{{\partial A}}{{\partial c}} \cdot \partial c \hfill \\ \frac{{\partial A}}{{\partial a}} = \frac{1}{2} \cdot \frac{{a{b^2} + a{c^2} - {a^3}}}{{\sqrt {2({a^2}{b^2} + {b^2}{c^2} + {c^2}{a^2}) - ({a^4} + {b^4} + {c^4})} }} = \frac{1}{2} \cdot \frac{{a \cdot ({b^2} + {c^2} - {a^2})}}{{\sqrt {(b + c - a)(a + b - c)(a - b + c)(a + b + c)} }} \hfill \\ \frac{{\partial A}}{{\partial a}}(10,17,21) = \frac{1}{2} \cdot \frac{{10 \cdot ({{17}^2} + {{21}^2} - {{10}^2})}}{{\sqrt {(17 + 21 - 10)(10 + 17 - 21)(10 - 17 + 21)(10 + 17 + 21)} }} = \frac{{75}}{8} \hfill \\ \frac{{\partial A}}{{\partial b}} = \frac{1}{2} \cdot \frac{{{a^2}b +
Answers Home All Categories Arts & Humanities Beauty & Style Business & Finance Cars & Transportation Computers & Internet Consumer Electronics Dining Out Education & Reference Entertainment & Music Environment Family & Relationships Food & Drink Games error propagation average & Recreation Health Home & Garden Local Businesses News & error propagation chemistry Events Pets Politics & Government Pregnancy & Parenting Science & Mathematics Social Science Society & error propagation log Culture Sports Travel Yahoo Products International Argentina Australia Brazil Canada France Germany India Indonesia Italy Malaysia Mexico New Zealand Philippines Quebec Singapore Taiwan Hong Kong Spain http://math.stackexchange.com/questions/1259263/area-of-triangle-and-uncertainty-estimation Thailand UK & Ireland Vietnam Espanol About About Answers Community Guidelines Leaderboard Knowledge Partners Points & Levels Blog Safety Tips Science & Mathematics Mathematics Next CALCULUS! differentials... use them to approximate the possible propagated error in computing the area ? The measurements of the base and altitude of a triangle are found https://answers.yahoo.com/question/?qid=20081030215018AAmOAoG to be 32 and 54 centimeters, respectively. The possible error in each measurement is 0.25 centimeter. Use differentials to approximate to one decimal place the possible propagated error in computing the area of the triangle. Update: I know that you have to take the derivative of the area formula... a=1/2 (bh) but.. how do you take a derivative of that with two constants? (b and h) Update 2: i have the answer... the derivative would be something like (1/4)b+(1/4)h *.25 which would give you 5.4... and for some reason you have to multiply that by two... or maybe if you just leave it at (1/2)b+(1/2)h... help. i just want to know.. its already past due... i just want to know. thanks! Update 3: actually yes... your derived formula would be... ((1/2)b+(1/2)h)*.25 and you get 10.75 and rounded to one decimal point you would get 10.8 haha. okay. but why is the equation
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