https://file.aiquickdraw.com/mathbot/file/1726612013098z973rt8t.jpg

The problem describes an electrical circuit that switches from a 6-volt battery to an 11-volt battery 3 seconds after being turned on. You are asked to define a piecewise function for the voltage \( V(t) \), depending on time \( t \).

Here’s how to approach the problem:

1. **Before the switch (for \( t < 3 \)):** The circuit uses the 6-volt battery, so:
   \[
   V(t) = 6 \quad \text{for} \quad t < 3
   \]

2. **After the switch (for \( t \geq 3 \)):** The circuit switches to the 11-volt battery, so:
   \[
   V(t) = 11 \quad \text{for} \quad t \geq 3
   \]

Thus, the voltage function \( V(t) \) can be written as:

\[
V(t) =
\begin{cases}
6 & \text{for} \ t < 3 \\
11 & \text{for} \ t \geq 3
\end{cases}
\]

### Discontinuity:

- The function experiences a discontinuity at \( t = 3 \), as the voltage jumps from 6 volts to 11 volts. Therefore, the function is discontinuous at \( t = 3 \).

### Final answers:

1. For \( t < 3 \), \( V(t) = 6 \).
2. For \( t \geq 3 \), \( V(t) = 11 \).
3. The function is discontinuous at \( t = 3 \).

---

Would you like a deeper explanation, or have any questions?

Here are 5 related questions to explore this topic further:

1. How is continuity of a function defined at a point?
2. What is the physical meaning of discontinuity in this electrical circuit example?
3. How would the graph of the voltage function \( V(t) \) look over time?
4. How could we modify the circuit so that the voltage changes continuously instead of discontinuously?
5. What are other real-world examples where functions might exhibit discontinuities?

**Tip:** In real-world systems, discontinuities in physical quantities like voltage often indicate instantaneous changes in state, such as switching between power sources.

An electrical circuit switches instantaneously from a 6-volt battery to an 11-volt battery 3 seconds after being turned on. Sketch on a sheet of paper a graph of the battery voltage against time. Then fill in the formulas below for the function represented by your graph. At what point or points is your function discontinuous?

Understand the behavior of a voltage function where an electrical circuit switches from a 6-volt battery to an 11-volt battery at t = 3 seconds. This problem covers piecewise functions, the concept of discontinuity, and voltage-time graph analysis. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation