Problem 48 - Elementary and Intermediate Algebra

Question

Discharging a Battery. The charge remaining in a battery decreases as the battery discharges. The charge \(C\) (in coulombs) after \(t\) days is given by the function \(C(t)=0.0003(0.7)^{t} .\) Find the charge after 5 days.

Step-by-Step Solution

Answer

The charge after 5 days is 0.000050421 coulombs.

Step 1: Understand the Function

The problem gives us the function for the charge remaining in a battery after \(t\) days: \(C(t) = 0.0003(0.7)^t\). This is an exponential decay function where the initial charge is scaled by \(0.7\) each day.

Step 2: Substitute the Given Value

We are asked to find the charge after 5 days, so we substitute \(t = 5\) into the function: \(C(5) = 0.0003(0.7)^5\).

Step 3: Calculate the Exponential Term

First, calculate \((0.7)^5\). This gives us \(0.7 \times 0.7 \times 0.7 \times 0.7 \times 0.7 = 0.16807\).

Step 4: Multiply by the Coefficient

Next, multiply this result by 0.0003: \(C(5) = 0.0003 \times 0.16807 = 0.000050421\).

Step 5: State the Final Charge

The final charge remaining in the battery after 5 days is \(0.000050421\) coulombs.

Key Concepts

Battery Discharge

Battery discharge is a common phenomenon where the stored energy in a battery depletes over time. This occurs as the battery is used, and its capacity to supply charge reduces. In contexts like electronic devices, understanding battery discharge is crucial because it affects the device's operational time.
The discharge process can be influenced by several factors, including:

• Temperature: Higher temperatures can increase the rate of discharge.
• Usage patterns: Continuous use leads to quicker discharge compared to occasional use.
• Battery type: Different battery technologies discharge at different rates.

In mathematical terms, battery discharge often follows an exponential decay model, which provides an accurate prediction of how quickly the charge diminishes over time.

Exponential Functions

Exponential functions are mathematical expressions that describe quantities changing at a consistent rate relative to their current value. These functions are characterized by a base raised to a power, which alters over time or through another variable.
In the context of battery discharge, the exponential function takes the form:\[C(t) = C_0(0.7)^t\]where:

• \(C(t)\) is the remaining charge at time \(t\).
• \(C_0\) is the initial charge.
• The base \(0.7\) indicates the decay factor, representing a 30% decrease in charge each day.

Exponential functions are powerful tools in modeling real-world processes due to their ability to represent growth or decay rates naturally. Whether you're observing how population grows, or how a battery depletes, exponential functions play a key role.

Mathematical Modeling

Mathematical modeling involves constructing mathematical formulas or functions to represent real-world situations. This approach helps visualize complex phenomena and predict future outcomes.
For battery discharge, mathematical modeling enables you to calculate future charge levels simply by altering the variable \(t\) in the function \(C(t) = C_0 (0.7)^t\). This model outlines how charge will diminish over time, letting users forecast when a battery might need recharging.
Models like this are invaluable for planning, such as ensuring you're not caught with a dead battery, or predicting device performance. By leveraging known initial conditions and decay rates, mathematical modeling presents a reliable way to comprehend and manage the discharge process efficiently.