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Weekly Quantitative Brain Teaser

A voltage, V volts, applied to a resistor of R ohms produces an electric current of I amps where V = IR. As the current flows the resistor heats up and its resistance falls. If 100 volts is applied to a resistor of 1000 ohms the current is initially 0.1 amps but rises by 0.001 amps/min. At what rate is the resistance falling if the voltage remains constant?

-From: Calculus: Single and Multivariable by Hughes-Hallett, Gleason, McCallum, et al.

*Come back next week for the answer!

Last Week's Brain Teaser

A spherical balloon is inflated so that its radius is increasing at a constant rate of 1 cm per second. At what rate is air being blown into the balloon when its radius is 5 cm? 

Answer:

Since we want the rate of air being blown into the balloon, we want to know the derivative of the volume with respect to time (cm³/s). Therefore, we need the equation for the volume of a sphere:
V(r) = (4/3)(pi)r³.
When taking the derivative, we need to remember that the radius depends on time, so what we really have is
V(r(t)) which requires use of the chain rule: dV(r(t))/dt = (dV/dr)(dr/dt).
dV/dr = 4(pi)r², so dV/dt = 4(pi)r²(dr/dt).
Now we can plug in that r = 5 cm, and dr/dt = 1 cm/s:
dV/dt = 4(pi)(5 cm)²(1 cm/s) = 100(pi) cm³/s

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