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The given problem requires calculating the amount of stored charge in the first 3 hours of operation for a device. The charge flow is described by the function \( F(t) = 10t^2 \sqrt{3} - t \), where \( F(t) \) is in units of electrical charge per minute, and \( t \) is the time in hours. The total amount of charge stored is obtained by integrating \( F(t) \) over the time interval from \( t = 0 \) to \( t = 3 \).

The approach involves the following steps:

1. **Interpretation**: We are asked to compute the integral of \( F(t) \) over the interval [0, 3], as this will give the total charge accumulated over the first 3 hours.

2. **Integration**: To solve this, we will integrate the function \( F(t) = 10t^2 \sqrt{3} - t \).

Let's compute this now.The result of the integration is \( -\frac{9}{2} + 90\sqrt{3} \).

Now, let's calculate the numerical value of this expression to determine the total amount of charge stored.The total amount of charge stored in the device during the 3 hours of operation is approximately \( 151.38 \) units of electrical charge.

Would you like more details on the steps, or have any other questions?

Here are five related questions to deepen your understanding:
1. How is the function \( F(t) \) derived in similar electrical systems?
2. What is the physical meaning of integrating the charge flow function?
3. How would the total charge change if the time period were extended?
4. Can this integration process be used in systems with variable voltage?
5. How does the square of time \( t^2 \) influence the charge accumulation?

**Tip**: Understanding the behavior of a function can be enhanced by plotting it over the time range. This gives insight into how quickly the charge accumulates.

Luego de realizar las pruebas técnicas se determinó que el flujo de carga que almacena un dispositivo desde que es conectado a la red eléctrica presenta un comportamiento según F(t) = 10t^2√3 - t. Además, el dispositivo solo puede estar conectado por 3 horas. Calcule la cantidad de carga almacenada en las 3 horas de funcionamiento.

Learn how to calculate the stored charge of a device using the function F(t) = 10t^2√3 - t over a 3-hour period. This problem demonstrates key concepts in integral calculus and electrical charge flow. Suitable for undergraduate engineering students.

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