answer:To find the distance from a point to a line, we can use the formula: d = frac{|ax_0 + by_0 + c|}{sqrt{a^2 + b^2}} where (x_0, y_0) is the point, and ax + by + c = 0 is the equation of the line. In this case, we have: a = 3, b = frac{1}{10}, c = frac{9}{10}, x_0 = -frac{16}{5}, y_0 = 4 Plugging these values into the formula, we get: d = frac{|3(-frac{16}{5}) + frac{1}{10}(4) + frac{9}{10}|}{sqrt{3^2 + (frac{1}{10})^2}} d = frac{|-frac{48}{5} + frac{2}{5} + frac{9}{10}|}{sqrt{9 + frac{1}{100}}} d = frac{|-frac{48}{5} + frac{2}{5} + frac{9}{10}|}{sqrt{frac{901}{100}}} d = frac{|-frac{48}{5} + frac{2}{5} + frac{9}{10}|}{frac{sqrt{901}}{10}} d = frac{frac{83}{10}}{frac{sqrt{901}}{10}} d = frac{83}{sqrt{901}} Therefore, the distance from the point left(-frac{16}{5}, 4right) to the line 3x+frac{y}{10}+frac{9}{10}=0 is frac{83}{sqrt{901}}. The answer is frac{83}{sqrt{901}}

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answer:The central theme of the passage is that Thomas Kinkade, known for his innovative technique of embedding messages in his art, is gaining significant popularity and may defy critics' expectations to become the most renowned artist of modern times.

answer:For the linear time-varying system dot x(t)=A(t)x(t) the solution is expressed as x(t)=Phi(t)x(0), with Phi(t)=expleft(int_0^t A(tau)dtauright) For the shifted system, frac{dx}{dt} = A(t+s)x(t) the corresponding fundamental matrix tilde{Phi}(t) can be derived as follows: begin{align*} tilde{Phi}(t) &= expleft(int_0^t A(tau+s)dtauright) &= expleft(int_s^{t+s} A(v)dvright) &= expleft(-int_0^s A(v)dv + int_0^{t+s} A(v)dvright) &= expleft(-int_0^s A(v)dvright) expleft(int_0^{t+s} A(v)dvright) &= Phi(s)^{-1} Phi(t+s) end{align*} Hence, the solution to the shifted system is given by x(t) = Phi(s)^{-1} Phi(t+s) x(0) To relate the Lyapunov exponents of the two systems, note that they are measures of the average rate of divergence or convergence of nearby trajectories. Since the systems are related through a time-shift and a similarity transformation, their Lyapunov exponents will be the same. However, the specific calculation of Lyapunov exponents for each system will depend on the original and shifted time-varying matrices A(t) and A(t+s).