Problem 82 - 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 maximum charge \(C_{0} ?\)

Step-by-Step Solution

Answer

It takes approximately 0.58 hours to charge the battery to 90% of its maximum charge.

Step 1: Understand the Formula

We are given the formula \(t=-k \ln \left(1-\frac{C}{C_{0}}\right)\) which relates time \(t\), charge \(C\), maximum charge \(C_{0}\), and a constant \(k\). Our task is to find time \(t\) when the battery is charged to 90\(\%\) of \(C_{0}\).

Step 2: Substitute Given Values

We know the battery needs to be charged to 90\(\%\) of its maximum charge \(C_{0}\), meaning \(C = 0.9 \times C_{0}\). The constant \(k\) is given as 0.25. Substitute these values into the formula: \[ t = -0.25 \ln \left(1-\frac{0.9C_{0}}{C_{0}}\right) \].

Step 3: Simplify the Expression Inside the Logarithm

Simplify \(1 - \frac{0.9C_{0}}{C_{0}}\) to obtain 0.1. The equation thus becomes \[ t = -0.25 \ln(0.1) \].

Step 4: Calculate the Natural Logarithm

Use a calculator to find the natural logarithm of 0.1, \( \ln(0.1) \approx -2.302 \).

Step 5: Calculate the Time \(t\)

Substitute \( \ln(0.1) \approx -2.302 \) back into the equation to find \( t \): \[ t = -0.25 \times -2.302 \]. Compute this to get \( t \approx 0.5755 \) hours.

Key Concepts

Natural Logarithm

The natural logarithm is a mathematical concept essential in various fields, particularly when dealing with growth and decay processes. It is denoted as \(\ln(x)\), which essentially means the logarithm to the base \(e\), where \(e\) is approximately 2.718.

• In our exercise, the natural logarithm helps model how the charge time changes as we approach the battery's full charge.
• This mathematical function is used because it smoothly and accurately represents processes that rise or fall at a rate proportional to their current value, such as battery charging.

When working with natural logarithms, it's important to remember a few properties:

• \(\ln(1) = 0\): This property indicates that any number raised to zero is one, meaning a fully charged battery would have a \(\ln\) value of zero once it's fully charged.
• \(\ln(ab) = \ln(a) + \ln(b)\): This property allows us to split the logarithm of a product, though it wasn't directly used here, it's good to know for solving other problems.
• \(\ln(a^b) = b\ln(a)\): This property shows that the logarithm of a power turns into a simple multiplication, making calculations such as natural logarithms more manageable.

These properties are pivotal when you work through calculations involving natural logarithms, as seen in the computation of battery charging time.

Battery Charging Time

The battery charging time formula given in the exercise demonstrates exponential decay characteristics. This formula links the time \(t\), the achieved charge \(C\), the maximum charge \(C_{0}\), and a specific constant \(k\).

• The variable \(t\) is what we calculate—the time taken to reach a particular charge level.
• The formula \(t = -k \ln\left(1 - \frac{C}{C_{0}}\right)\) predicts that charging becomes slower as the battery approaches its full capacity \(C_{0}\).
• This happens because the rate of charging decreases exponentially, making it harder to fill up the last remaining percentage of the battery.

With a constant \(k\) value of 0.25 given in the exercise, the calculation becomes straightforward:

• For charging to 90% of \(C_{0}\), we set \(C = 0.9 \times C_{0}\).
• This translates to calculating \(t = -0.25 \ln(0.1)\), simplifying to \(t \approx 0.5755\) hours.

It shows that the final timescale is not a linear function of the charge percent but one that follows a natural logarithm, demonstrating why exponential decay is an excellent model for charging behavior.

Maximum Charge

The concept of maximum charge \(C_{0}\) is vital when discussing battery charging and the exponential decay relationship it shares with charging time.

• The maximum charge \(C_{0}\) represents the full capacity of the battery, or 100% charge, after which no more energy can be stored.
• In our exercise, this amount never changes, meaning it's a fixed point when analyzing how close the battery is to being fully charged.

This act of charging isn't as straightforward as merely adding energy at a constant rate. Instead, the closer the battery gets to \(C_{0}\), the more energy input is required to increase the charge, as represented by the fractional term \(1 - \frac{C}{C_{0}}\) in the formula.In practice, understanding the maximum charge is crucial for efficient battery usage:

• It helps determine how long you can use a fully charged battery before needing a recharge.
• It optimizes use, ensuring you charge only up to the necessary level, preserving battery health and longevity.

Grasping the idea of \(C_{0}\) prepares you to manage real-world battery systems better, whether it be for your smartphone or electric vehicle.