Problem 90 - Precalculus Mathematics for Calculus

Question

Charging a Battery The rate at which a battery charges is slower the closer the battery is to its maximum charge \(C_{0}\). The time (in hours) required to charge a fully discharged battery to a charge \(C\) is given by $$t=-k \ln \left(1-\frac{C}{C_{0}}\right)$$ where \(k\) is a positive constant that depends on the battery. For a certain battery, \(k=0.25 .\) If this battery is fully discharged, how long will it take to charge to \(90 \%\) of its maximum charge \(C_{0} ?\)

Step-by-Step Solution

Answer

It takes approximately 0.576 hours to charge to 90% of its maximum.

Step 1: Understand the problem

We need to find the time it takes to charge a fully discharged battery to 90% of its maximum charge using the given formula. Here, we identify the key values: the charge to achieve is \( C = 0.9C_0 \) and \( k = 0.25 \).

Step 2: Set up the equation

Substitute \( C = 0.9C_0 \) into the time equation: \[ t = -k \ln \left(1 - \frac{C}{C_{0}}\right) \Rightarrow t = -0.25 \ln \left(1 - \frac{0.9C_{0}}{C_{0}}\right) \]

Step 3: Simplify the equation

Simplify the expression inside the logarithm: \[ 1 - \frac{0.9C_0}{C_0} = 1 - 0.9 = 0.1 \] So the equation becomes: \[ t = -0.25 \ln(0.1) \]

Step 4: Calculate the logarithm value

Compute \( \ln(0.1) \). Using a calculator, we find \[ \ln(0.1) \approx -2.302 \]

Step 5: Compute the total time

Substitute \( \ln(0.1) \approx -2.302 \) back into the equation: \[ t = -0.25 (-2.302) \] Calculate the product: \[ t = 0.5755 \] hours.

Key Concepts

Logarithmic Functions

Logarithmic functions are mathematical functions that are the inverse of exponential functions. In simpler terms, if an exponential function can be expressed as \( y = a^x \), then its corresponding logarithmic function is \( x = \log_a(y) \). Logs can help solve equations involving exponential growth or decay, which is common in battery charging scenarios.
Logarithmic calculations are essential for understanding many real-world processes because they can transform multiplicative processes into additive ones, making them easier to handle. For example, when determining the time to charge a battery, as in our exercise, we're essentially converting the complex exponential process of charging into something more manageable using logarithms. This way, the rate of change gets expressed clearly, and we can calculate it straightforwardly.
When solving problems like our battery charging scenario, understanding logs enable us to isolate the time variable, making the calculation much more accessible and straightforward.

Battery Charging

Battery charging is an example of exponential growth and decay, where the rate of energy stored in a battery decreases as it approaches its maximum charge capacity, \( C_0 \).
During the charging process, initially, the current flows quickly into the battery, charging it rapidly. However, as it gets closer to being fully charged, the rate slows down significantly, following an exponential pattern. This is why the charging time increases as the battery is almost full.

• Fully Discharged: Starts charging quickly, rapid initial charge.
• Closer to Full Capacity: Slowdown occurs, charging rate decreases.

This behavior is captured in the logarithmic function used in the problem. The logarithm here helps model this behavior very accurately by demonstrating how charging becomes slower in proportion to how full the battery is.

Rate of Change

Rate of change is crucial when discussing exponential growth or decay, such as in battery charging. It tells us how quickly a quantity is changing over a certain period. In this context, it's the speed at which the battery charges over time.
We used a specific formula \( t = -k \ln(1-\frac{C}{C_0}) \) to describe battery charging time. Here, the negative sign is significant, showing reversed exponential behavior – indicating slower change as time progresses.
This demonstrates that:

• The rate of change is initially high at the start of the charging and diminishes over time.
• The constant \( k \) scales the time calculation, representing the intrinsic properties of the battery that affect how fast it charges.

Understanding the rate of change is essential for grasping how quickly resources like energy are used or replenished.

Time Calculation

Calculating time in scenarios involving logarithmic functions often involves understanding both the process being described and the mathematical operations involved.
In our specific battery-charging problem, the time calculation involves substituting values into the logarithm-based formula.
The formula \( t = -0.25 \ln(0.1) \) gives the charging time for 90% capacity by simplifying the input of the logarithm.

• Begin by substituting known values such as the fraction of maximum charge intended (90% in our case).
• Simplify the expression inside the logarithm to make it more straightforward to calculate.
• Compute the logarithm, then multiply by \( -k \) to find the required time.

It's essential to use accurate calculation tools or software to find logarithm values precisely, as this influences the final result. The ultimate goal is finding how long the process, like charging, will take given specific conditions.