Volume and its errorโฆ Volume, VV = 40′ ร 20′ ร 15′ = 12,000 ft3 Take partial derivativesโฆ โVโL=WH\frac{\partial V}{\partial L}=WH, โVโW=LH\frac{\partial V}{\partial W}=LH, and โVโH=LW\frac{\partial V}{\partial H}=LW. Now, SV=(โVโLSL)2+(โVโWSW)2+(โVโHSH)2S_V=\sqrt{\left(\frac{\partial V}{\partial L}S_L\right)^2+\left(\frac{\partial V}{\partial W}S_W\right)^2+\left(\frac{\partial V}{\partial H}S_H\right)^2}. Now, only thing left to do is fill in the variables. Which is =(WH)2(0.05)2+(LH)2(0.03)2+(LW)2(0.02)2=\sqrt{(WH)^2(0.05)^2+(LH)^2(0.03)^2+(LW)^2(0.02)^2};=((20โ 15)ร0.05)2+((40โ 15)ร0.03)2+((40โ 20)ร0.02)2=\sqrt{((20\cdot15)\times0.05)^2+((40\cdot15)\times0.03)^2+((40\cdot20)\times0.02)^2}; =225+324+256=805=\sqrt{225+324+256}=\sqrt{805}= ยฑ 28 ft3
ฯx1x2\sigma_{x_1x_2} is known as the covariance between x1x_1 and x2x_2 For a function with n independent variables, the errors propagate as: ฮฃzz=[a1a2โฏan][ฯx12ฯx1x2โฏฯx1xnฯx2x1ฯx22โฏฯx2xnโฎโฎโฑโฎฯxnx1ฯxnx2โฏฯxn2][a1a2โฎan]\Sigma_{zz}= \left[ \begin{matrix} a_1 & a_2 & \cdots & a_n \end{matrix} \right] \left[ \begin{matrix} \sigma_{x_1}^{2} & \sigma_{x_1x_2} & \cdots & \sigma_{x_1x_n}\\ \sigma_{x_2x_1} & \sigma_{x_2}^{2} & \cdots & \sigma_{x_2x_n}\\ \vdots & \vdots & \ddots &…
Jacobian Matrix set-up: J=[โFโxโFโyโGโxโGโy]J=\begin{bmatrix} \dfrac{\partial F}{\partial x} & \dfrac{\partial F}{\partial y} \\ \dfrac{\partial G}{\partial x} & \dfrac{\partial G}{\partial y}\end{bmatrix} , where JJ is the coefficient of linearized equations. For this example let’s take the two equations F(x,y)=x+yโ2y2=4F(x,y)=x+y-2y^2=4 and G(x,y)=x2+y2=8G(x,y)=x^2+y^2=8. (Use x0=1x_0=1 and y0=1y_0=1 for initial approximations.) Let’s do the partial derivative calculate to plug into…