Optimizing Solar Panel Power Output: Calculating dP/dt and θ'(t)

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In summary, the power output of a solar panel is dependent on the angle of the sun shining on it, with P(θ) representing the output for an angle θ between 0 and π. The angle θ(t) between the panel and the sun at a given time t on a typical summer day in Ann Arbor can be calculated using the function θ(t) = arcsin(t/7 -1) + π/2. Using the chain rule, the average rate of change of power with respect to time is given by dP/dt = P'(θ(t)) * θ'(t), where P'(θ(t)) is the average rate of change of power with respect to θ and θ'(t
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IntegrateMe
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The power output of a particular type of solar panel varies with the angle of the sun shining on the panel. The panel outputs P (θ) watts when the angle between the sun and the panel is θ for 0 ≤ θ ≤ π. On a typical summer day in Ann Arbor, the angle between a properly mounted panel and the sun t hours after 6 a.m. is θ(t) for 0 ≤ t ≤ 14. Assume that sunrise is at 6 a.m. and sunset is 8 p.m.

(A) Calculate dP/dt using the chain rule, and give interpretations for each part of your calculation.

P(θ(t)). So, dP/dt = P(θ(t))' = P'(θ(t)) * θ'(t)
P'(θ(t)) is the average rate of change of power with respect to θ.
θ'(t) is the average rate of change of θ with respect to t.


(B) Suppose θ(t) = arcsin(t/7 -1) + π/2. Calculate θ'(t) using the equivalent expression: sin(θ(t) - π/2) = t/7 - 1

I just differentiated the equivalent function:

cos(θ(t) - π/2)*θ'(t) = 1/7
θ'(t) = 1/(7*cos(θ(t) - π/2))

(C) Suppose dP/dθ (2π/3) = 12 and θ(t) is the function in part (B). Find the change in power output between 4:30PM and 5:30PM.

This is where I'm having trouble. I would think the "change in power output" would simply be dP/dθ, since this represents the change in power with respect to θ, but I feel as though I'm incorrect here. Any help would be awesome. Thanks!
 
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This is where I'm having trouble. I would think the "change in power output" would simply be dP/dθ, since this represents the change in power with respect to θ, but I feel as though I'm incorrect here. Any help would be awesome. Thanks!

I don't know what they mean by the change in power output. Probably it's P(5:30 PM) - P(4:30 PM), since a change is just a variation of a function. In that case you have to find P(t), integrating the expression in (A).

Sorry if this didn't help :tongue2:
 

What is the purpose of optimizing solar panel power output?

The purpose of optimizing solar panel power output is to maximize the efficiency and energy production of solar panels, ultimately reducing the cost of energy and promoting the use of renewable energy sources.

What is dP/dt and how is it calculated?

dP/dt, also known as the power rate of change, is the instantaneous rate of change of power output with respect to time. It can be calculated by taking the derivative of the power output function with respect to time.

Why is it important to calculate dP/dt for solar panels?

Calculating dP/dt allows us to understand the rate at which the power output of solar panels is changing, which is crucial for optimizing their efficiency. It also helps us identify any potential issues or limitations that may be affecting the performance of the panels.

What is θ'(t) and how is it related to solar panel power output?

θ'(t), also known as the angle of incidence, is the rate of change of the angle at which sunlight hits the solar panel. This angle has a direct impact on the amount of energy that can be converted by the panel, making it an important factor to consider when optimizing power output.

What are some methods for optimizing solar panel power output based on dP/dt and θ'(t)?

Some methods for optimizing solar panel power output include adjusting the tilt and orientation of the panels to maximize sunlight exposure, using tracking systems to follow the sun's movement, and implementing cleaning and maintenance routines to ensure the panels are functioning at their best.

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