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We replace both occurrences of e −k t with Q: This is now a fractional equation in Q. In order to clear the denominators, multiply both sides by 1 − Q: a (1 − Q) = b (1 + Q). This is now a linear equation in Q. In order to collect like terms, distribute on both sides: a − a Q = b + b Q. Collect the constant terms on the left-hand-side.

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Here the variables are the radius $r$ and the volume $V$. We know $dV/dt$, and we want $dr/dt$. The two variables are related by means of the equation $\ds V=4\pi r^3/3$. Taking the derivative of both sides gives $\ds dV/dt=4\pi r^2\dot r$.

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The common transposition method is to do the same thing (mathematically) to both sides of the equation, with the aim of bringing like terms together and isolate the variable (or the unknown quantity). For example: 5x + 8 = 3x – 6. We want to get rid of the number 8 from the left side. So we subtract 8 from both sides of the equation.

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Steel Beam Calculator Ltd was set up in 2011 to enable building industry professionals to design safe and effective steel beams more quickly and affordably. This calculator was built to design efficient steel beams using the most advanced Eurocodes.

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This is actually two calculators; the upper doing the math calculations for the length of the two undetermined sides and the undetermined angle of the simulated Stimpmeter, visualized as a triangle. To use this calculator, verify that side c has a value in it and angles A and C have values in them; the default values present should be correct ...

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The combined area of sets A and B is the union of A and B, as we discussed in the previous section. The union in this case contains all living creatures that are either two-legged or that can fly or both. Moreover, the area in both A and B, where the two sets overlap, is called intersection of A and B, as discussed in the previous section.

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Square both sides first to get rid of the outside square root. Then simplify, get the radical by itself on one side, and square both sides again. Check both answers to make sure we have no negatives under radicals. Note that \(\displaystyle -\frac{{\sqrt{3}}}{2}\) makes the outside radical negative, so it is extraneous.