Question

The amount of electricity that a solar panel is capable of producing slowly decays over time. After ten years, a solar panel produces 89% of the electricity that it was able to produce when it was brand new. Find the exponential decay constant k. If a solar panel is initially capable of producing 450 watts of power, how long will it take before the solar panel is only able to produce 300 watts of power?

Answer 1

Answer:

It will take 34.79 years before the solar panel is only able to produce 300 watts of power

Step-by-step explanation:

The equation for the amount of electricity that a solar panel is capable has the following format:

$Q(t) = Q(0)e^{-kt}$

In which Q(t) is the amount after t years, Q(0) is the initial amount and k is the exponential decay constant.

After ten years, a solar panel produces 89% of the electricity that it was able to produce when it was brand new.

This means that $Q(10) = 0.89Q(0)$. So

$Q(t) = Q(0)e^{-kt}$

$0.89Q(0) = Q(0)e^{-10k}$

$e^{-10k} = 0.89$

$\ln{e^{-10k}} = \ln{0.89}$

$-10k = \ln{0.89}$

$10k = -\ln{0.89}$

$k = \frac{-\ln{0.89}}{10}$

$k = 0.01165$

So

$Q(t) = Q(0)e^{-0.01165t}$

Initially capable of producing 450 watts of power

This means that $Q(0) = 450$

How long will it take before the solar panel is only able to produce 300 watts of power?

This is t for which Q(t) = 300. So

$Q(t) = 450e^{-0.01165t}$

$450 = 300e^{-0.01165t}$

$e^{-0.01165t} = \frac{300}{450}$

$\ln{e^{-0.01165t}} = \ln{\frac{300}{450}}$

$-0.01165t = \ln{\frac{300}{450}}$

$0.01165t = -\ln{\frac{300}{450}}$

$t = -\frac{\ln{\frac{300}{450}}}{0.01165}$

$t = 34.79$

It will take 34.79 years before the solar panel is only able to produce 300 watts of power