Problem 90 - Algebra and Trigonometry

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 maxi- mum charge \(C_{0} ?\)

Step-by-Step Solution

Answer

It takes approximately 0.5755 hours to charge the battery to 90\% of its maximum capacity.

Step 1: Identify Known Values

In the given problem, we know that the constant \(k = 0.25\). We are asked how long it takes to charge a battery to 90\(\%\) of its maximum charge \(C_0\). Therefore, \(C = 0.9C_0\).

Step 2: Substitute Values into the Formula

The formula provided is \( t = -k \ln\left(1 - \frac{C}{C_0}\right) \). Substitute \(k = 0.25\) and \(C = 0.9C_0\) into the formula to get: \[ t = -0.25 \ln\left(1 - \frac{0.9C_0}{C_0}\right) \] which simplifies to \[ t = -0.25 \ln(1 - 0.9) \]

Step 3: Simplify the Expression Inside the Logarithm

Compute the expression inside the logarithm: \( 1 - 0.9 = 0.1 \). Substitute back into the formula, we have: \[ t = -0.25 \ln(0.1) \]

Step 4: Calculate the Natural Logarithm

Calculate the natural logarithm of 0.1, which is \( \ln(0.1) \approx -2.302 \).

Step 5: Compute the Charging Time

Substitute \( \ln(0.1) = -2.302 \) into the equation: \[ t = -0.25 \times (-2.302) \] Calculate \( t \): \[ t = 0.5755 \]

Step 6: Interpret the Result

The value of \( t \approx 0.5755 \) hours. This means it will take approximately 0.5755 hours to charge the battery to 90\% of its maximum capacity.

Key Concepts

Exponential Function

The concept of an exponential function is fundamental in understanding how battery charging works. An exponential function can be recognized by its constant multiplier on a variable exponent, specifically in the form \( a^x \), where \( a \) is a constant and \( x \) is the variable. In the context of battery charging, this function helps describe the non-linear relationship between the charging time and the remaining capacity of the battery.
As a battery approaches its full charge, the rate of charging slows down exponentially. This means that while the initial portion of the battery charges quickly, the final stretch takes significantly longer. This model is reflective of real-world charging habits and helps engineers design more efficient charging protocols.

Natural Logarithm

The natural logarithm, indicated as \( \,\ln(x) \, \), is crucial for our calculations since it's the inverse operation of the exponential function with base \( e \), which is approximately equal to 2.718. The natural logarithm of a number is essentially the power to which \( e \) must be raised to obtain that number.
By understanding \( \ln \), we can manipulate equations and solve for unknowns, like in the battery charge formula, where \( \ln \left(1 - \frac{C}{C_0}\right) \) helps us determine the balance between charged and uncharged capacity. The nature of logarithms, being non-linear, aligns well with the exponential decay seen in battery charging rates.

Charging Efficiency

Charging efficiency is a measure of how effectively a battery can convert the energy from a charger into stored charge. Not all of the energy from the charger gets stored due to various factors like heat dissipation and internal resistance.
Understanding charging efficiency involves looking at the real-world performance of a battery. It's crucial to optimize this efficiency to ensure more energy is stored in less time, reducing wastage. While high charging efficiency is desirable, different battery technologies and designs will have varying efficiency levels. For instance, fast-charging technologies try to optimize this efficiency, but often face the challenge of increased heat.

Battery Charging Formula

The battery charging formula \( t = -k \ln \left(1-\frac{C}{C_0}\right) \) gives us a mathematical model to calculate the time needed for charging. This formula incorporates several key elements: the constant \( k \), the natural logarithm, and the ratio of current to maximum charge, \( \frac{C}{C_0} \).
The constant \( k \) is specific to each battery system and reflects how quickly a battery can accept charge over time. In practice, this formula acknowledges that as a battery reaches closer to its full capacity, it becomes significantly slower to charge. By substituting the known values into this formula, we can predict and plan charging sessions more effectively, maximizing battery life and performance.