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 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:
$$\begin{gathered} A = \frac{1}{4}\sqrt {2({a^2}{b^2} + {b^2}{c^2} + {c^2}{a^2}) - ({a^4} + {b^4} + {c^4})} ;\,\,\,\,\,\partial A = \frac{{\partial A}}{{\partial a}} \cdot \partial a + \frac{{\partial A}}{{\partial b}} \cdot \partial b + \frac{{\partial A}}{{\partial c}} \cdot \partial c \hfill \\ \frac{{\partial A}}{{\partial a}} = \frac{1}{2} \cdot \frac{{a{b^2} + a{c^2} - {a^3}}}{{\sqrt {2({a^2}{b^2} + {b^2}{c^2} + {c^2}{a^2}) - ({a^4} + {b^4} + {c^4})} }} = \frac{1}{2} \cdot \frac{{a \cdot ({b^2} + {c^2} - {a^2})}}{{\sqrt {(b + c - a)(a + b - c)(a - b + c)(a + b + c)} }} \hfill \\ \frac{{\partial A}}{{\partial a}}(10,17,21) = \frac{1}{2} \cdot \frac{{10 \cdot ({{17}^2} + {{21}^2} - {{10}^2})}}{{\sqrt {(17 + 21 - 10)(10 + 17 - 21)(10 - 17 + 21)(10 + 17 + 21)} }} = \frac{{75}}{8} \hfill \\ \frac{{\partial A}}{{\partial b}} = \frac{1}{2} \cdot \frac{{{a^2}b + b{c^2} - {b^3}}}{{\sqrt {2({a^2}{b^2} + {b^2}{c^2} + {c^2}{a^2}) - ({a^4} + {b^4} + {c^4})} }} = \frac{1}{2} \cdot \frac{{b \cdot ({a^2} + {c^2} - {b^2})}}{{\sqrt {(b + c - a)(a + b - c)(a - b + c)(a + b + c)} }} \hfill \\ \frac{{\partial A}}{{\partial b}}(10,17,21) = \frac{1}{2} \cdot \frac{{17 \cdot ({{10}^2} + {{21}^2} - {{17}^2})}}{{\sqrt {(17 + 21 - 10)(10 + 17 - 21)(10 - 17 + 21)(10 + 17 + 21)} }} = \frac{{51}}{8} \hfill \\ \frac{{\partial A}}{{\partial c}} = \frac{1}{2} \cdot \frac{{{b^2}c + {a^2}c - {c^3}}}{{\sqrt {2({a^2}{b^2} + {b^2}{c^2} + {c^2}{a^2}) - ({a^4} + {b^4} + {c^4})} }} = \frac{1}{2} \cdot \frac{{c \cdot ({a^2} + {b^2} - {c^2})}}{{\sqrt {(b + c - a)(a + b - c)(a - b + c)(a + b + c)} }} \hfill \\ \frac{{\partial A}}{{\partial c}}(10,17,21) = \frac{1}{2} \cdot \frac{{21 \cdot ({{10}^2} + {{17}^2} - {{21}^2})}}{{\sqrt {(17 + 21 - 10)(10 + 17 - 21)(10 - 17 + 21)(10 + 17 + 21)} }} = - \frac{{13}}{8} \hfill \\ \partial A = \frac{{75}}{8} \cdot ( \pm 0.1) + \frac{{51}}{8} \cdot ( \pm 0.3) - \frac{{13}}{8} \cdot ( \pm 0.4) = \pm 2.2\,{{\text{m}}^2} \hfill \\ \end{gathered} $$
$$\left| {\frac{{\partial A}}{A}} \right| = \left| {\frac{{ \pm 2.2\,{{\text{m}}^2}}}{{84\,{{\text{m}}^2}}}} \right| \le \frac{{11}}{{420}} \approx 2.62\% $$
Remark:
What is meant by the phrase "approximate upper limit for the uncertainty in the approximation of area"? 1) Propagated error in calculating the area? 2) Relative error? 3) Relative error in percentage? The prof. used the phrase that is not commonly used. Hence I asked this question here to see how would people interpret this phrase. I have solved this problem but not sure if I did it right.
