```Question 127357
I hope that I correctly interpret your two given equations (one for gain and the other for
sensitivity) as being:
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{{{G = (B*t)/(R + R[t])}}}
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For this one I'm guessing on what the right side of this equation should be. You wrote it as:
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{{{G = (B*t)/R + R[t]}}}
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but I'm guessing that my version above is you intended. [If I'm wrong, you may want to repost your
problem and we'll try again.]
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and
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{{{S = (B*R)/(R + R[t])^2}}}
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You are given that B = 3.7, t = 90, G = 0.4, S = 0.001
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Substitute these values into the two equations and they become:
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{{{0.4 = (3.7*90)/(R + R[t])}}} <=== call this equation #1
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and
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{{{0.001 = (3.7*R)/(R + R[t])^2}}} <=== call this equation #2
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Go to equation #1 and multiply both sides by the quantity {{{R + R[t]}}}. When you do,
on the right side this multiplication cancels the denominator and you are left with equation #1
becoming:
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{{{0.4*(R + R[t]) = 3.7*90}}}
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Multiply out the right side and you have:
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{{{0.4*(R + R[t]) = 333}}}
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Finally solve for {{{R + R[t]}}} by dividing both sides of this equation by 0.4 to get:
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{{{R + R[t] = 333/0.4 = 832.5}}}
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You now have a numerical value of 832.5 ohms for R + R[t] and you can substitute that value in
for the denominator in equation #2 to make equation #2 become:
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{{{0.001 = (3.7*R)/(832.5)^2}}}
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Square out the denominator on the right side and the equation becomes:
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{{{0.001 = (3.7*R)/693056.25}}}
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Get rid of the denominator on the right side by multiplying both sides of this equation by
693056.25 and you get:
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{{{0.001*693056.25 = 3.7*R}}}
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Multiply out the left side and you reduce the equation to
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{{{693.05625 = 3.7*R}}}
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Finally solve for R by dividing both sides by 3.7 and you have:
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{{{187.3125 = R}}}
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[If only resistors could be manufactured to that precision ...]
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Now that we have a value for {{{R}}} we can return to either of the original equations  and substitute
this value of {{{R}}} and solve the resulting equation for {{{R[t]}}}.
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Let's return to equation #1 and set {{{R = 187.3125}}}. This makes equation #1 become:
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{{{0.4 = (3.7*90)/(187.3125 + R[t])}}}
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Again multiply out the numerator on the right side and you have:
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{{{0.4 = 333/(187.3125 + R[t])}}}
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Get rid of the denominator on the right side of this equation by multiplying both sides
by {{{187.3125 + R[t]}}} and the equation then is changed to:
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{{{0.4(187.3125 + R[t]) = 333}}}
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Get rid of the multiplier 0.4 by dividing both sides of the equation by 0.4 and the equation
is reduced to:
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{{{187.3125 + R[t] = 832.5}}}
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Get rid of the constant on the left side by subtracting 187.3125 from both sides and you are