5
advanced-math

A superconductor's resistance drops as R(T)=50e0.1TR(T) = 50e^{-0.1T} ohms where TT is temperature in Kelvin. At what temperature is resistance 5 ohms?

A

About 23 K

B

About 30 K

C

About 15 K

D

About 40 K

Correct Answer: A

Choice A is the correct answer. Solve for TT when R(T)=5R(T) = 5.

  1. Equation: 5=50e0.1T5 = 50e^{-0.1T}.
  2. Divide: 0.1=e0.1T0.1 = e^{-0.1T}.
  3. Natural log: ln(0.1)=0.1T\ln(0.1) = -0.1T.
  4. Solve: T=ln(0.1)0.1=2.3030.123.0323T = \frac{\ln(0.1)}{-0.1} = \frac{-2.303}{-0.1} \approx 23.03 \approx 23 K.

💡 Strategic Tip:ln(0.1)=ln(10)2.303\ln(0.1) = -\ln(10) \approx -2.303.

Choice B is incorrect because this is too high. Choice C is incorrect because this is too low. Choice D is incorrect because this is way too high.