Problem 78
Question
When a camera flash goes off, the batteries immediately begin to recharge the flash's capacitor, which stores electric charge given by $$ Q(t)=Q_{0}\left(1-e^{-t / a}\right) $$ (The maximum charge capacity is \(Q_{0}\) and \(t\) is measured in seconds.) (a) Find the inverse of this function and explain its mcaning. (b) How long does it take to recharge the capacitor to 90\(\%\) of capacity if \(a=2 ?\)
Step-by-Step Solution
Verified Answer
Inverse: \(t(Q) = -a \ln(1 - \frac{Q}{Q_0})\), 4.604 seconds for 90% recharge.
1Step 1: Understanding the Given Function
The function given is \( Q(t) = Q_0 (1 - e^{-t/a}) \). Here, \(Q(t)\) represents the electric charge at time \(t\), \(Q_0\) is the maximum charge capacity, \(a\) is a constant, and \(e\) is the base of the natural logarithm.
2Step 2: Finding the Inverse Function
To find the inverse, we solve for \(t\) in terms of \(Q\). Set \(Q(t) = y\) so \(y = Q_0 (1 - e^{-t/a})\). Rearrange this to find \(e^{-t/a}\):1. Divide both sides by \(Q_0\): \( \frac{y}{Q_0} = 1 - e^{-t/a} \).2. Rearrange to \( e^{-t/a} = 1 - \frac{y}{Q_0} \).3. Take the natural logarithm of both sides: \( -\frac{t}{a} = \ln(1 - \frac{y}{Q_0}) \).4. Multiply by \(-a\) to isolate \(t\): \( t = -a \ln(1 - \frac{y}{Q_0}) \).The inverse function is \( t(Q) = -a \ln(1 - \frac{Q}{Q_0}) \). This function gives the time \(t\) needed to reach a specific charge \(Q\).
3Step 3: Solving for Time to Reach 90% of Capacity
Substitute \(Q = 0.9Q_0\) into the inverse formula to find the time for 90% charge:1. Use the inverse function: \( t = -a \ln(1 - \frac{0.9Q_0}{Q_0}) \).2. Simplify: \( t = -a \ln(0.1) \).3. Substituting \(a = 2\): \( t = -2 \ln(0.1) \).4. Calculate: \( \ln(0.1) \approx -2.302 \), so: \( t = -2(-2.302) \approx 4.604 \text{ seconds} \).It takes approximately 4.604 seconds to recharge to 90% of capacity.
Key Concepts
Exponential FunctionsNatural LogarithmMathematical Modeling
Exponential Functions
Exponential functions are a type of mathematical function where the variable appears as an exponent. They are often written in the form \( f(x) = b^x \), where \( b \) is a constant greater than zero, and \( x \) is the exponent or power. These functions are recognized for their characteristic pattern of rapid growth or decay, depending on the value of the base \( b \).
In the context of our exercise, the function \( Q(t) = Q_0 \left(1 - e^{-t/a}\right) \) is not a pure exponential function because it involves a subtraction from one, but it does incorporate an exponential decay component \( e^{-t/a} \).
Exponential functions are used in a variety of real-world applications such as population growth modeling, radioactive decay, and compounding interest. They help to describe processes where the rate of change is proportional to the current amount, which is why they are especially useful in mathematical modeling scenarios where exponential growth or decay is evident.
When dealing with exponential functions, it's important to understand their properties:
In the context of our exercise, the function \( Q(t) = Q_0 \left(1 - e^{-t/a}\right) \) is not a pure exponential function because it involves a subtraction from one, but it does incorporate an exponential decay component \( e^{-t/a} \).
Exponential functions are used in a variety of real-world applications such as population growth modeling, radioactive decay, and compounding interest. They help to describe processes where the rate of change is proportional to the current amount, which is why they are especially useful in mathematical modeling scenarios where exponential growth or decay is evident.
When dealing with exponential functions, it's important to understand their properties:
- They are continuous and smooth.
- They have a horizontal asymptote, typically along the x-axis for decay functions.
- Their range is always positive when the base \( b > 0 \).
Natural Logarithm
The natural logarithm is a special logarithm whose base is the irrational number \( e \), approximately equal to 2.71828. It is often denoted as \( \ln(x) \). The natural logarithm is the inverse function of the exponential function involving \( e \). This means \( \ln(e^x) = x \) and \( e^{\ln(x)} = x \).
In our problem, we utilize the natural logarithm to find the inverse of the function \( Q(t) = Q_0 \left(1 - e^{-t/a}\right) \). To solve for the time variable \( t \) in terms of charge \( Q \), we first rearrange the given function and then apply the natural logarithm. When finding the inverse function, we arrive at: \( t(Q) = -a \ln\left(1 - \frac{Q}{Q_0}\right) \).
The use of the natural logarithm here allows us to handle exponential decay and extract time \( t \) from the exponent. The natural logarithm scales Beni to linearize the exponential curve, providing a straightforward method to derive \( t \). This is particularly useful as it breaks down complex growth or decay into a manageable form, shining light on just how quickly or slowly these processes occur.
In our problem, we utilize the natural logarithm to find the inverse of the function \( Q(t) = Q_0 \left(1 - e^{-t/a}\right) \). To solve for the time variable \( t \) in terms of charge \( Q \), we first rearrange the given function and then apply the natural logarithm. When finding the inverse function, we arrive at: \( t(Q) = -a \ln\left(1 - \frac{Q}{Q_0}\right) \).
The use of the natural logarithm here allows us to handle exponential decay and extract time \( t \) from the exponent. The natural logarithm scales Beni to linearize the exponential curve, providing a straightforward method to derive \( t \). This is particularly useful as it breaks down complex growth or decay into a manageable form, shining light on just how quickly or slowly these processes occur.
Mathematical Modeling
Mathematical modeling is a powerful way to describe real-world phenomena through mathematical expressions and equations. It involves creating representations of systems using mathematical language, allowing predictions and insights into their behavior.
In this exercise, mathematical modeling is evident in the representation of the electric charge stored in a camera's flash capacitor over time. The function \( Q(t) = Q_0 \left(1 - e^{-t/a}\right) \) models the relationship between time \( t \) and charge \( Q \).
This type of modeling is particularly useful because:
In this exercise, mathematical modeling is evident in the representation of the electric charge stored in a camera's flash capacitor over time. The function \( Q(t) = Q_0 \left(1 - e^{-t/a}\right) \) models the relationship between time \( t \) and charge \( Q \).
This type of modeling is particularly useful because:
- It helps in understanding and predicting the behavior of the capacitor during the recharging process.
- It enables us to calculate specific values, such as the time needed to reach a certain percentage of maximum capacity, enhancing our practical understanding of electronics.
- Such models can be adapted to similar scenarios, like the discharge of energy in other devices.
Other exercises in this chapter
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