It's fairly common in electrical circuits for additional, undesirable resistances to occur because of factors such as dirty, corroded, or loose connections. Suppose that a device with resistance R normally dissipates power P, but due to an additional series resistance r, the total power is reduced to P'. We might, for example, detect this change because the battery powering our device ran down more slowly than normal. (a) Find the unknown resistance r in terms of P, P', and R. (b) Check that the units of your result make sense. (c) Check that your result makes sense in the special cases P' = P and P' = 0. (d) Suppose we redefine P' as the useful power dissipated in R. For example, this would be the change we would notice because a flashlight was dimmer. Find r.

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