Problem 48 - 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 approximately 0.000050421 coulombs.

Step 1: Understand the given function

The function provided is \(C(t) = 0.0003(0.7)^t\). It represents the charge \(C\) in coulombs remaining in the battery after \(t\) days.

Step 2: Substitute the value of \(t\)

To find the charge after 5 days, we substitute \(t = 5\) into the function: \(C(5) = 0.0003(0.7)^5\).

Step 3: Calculate \((0.7)^5\)

Calculate the value of \((0.7)^5\). First, calculate \((0.7)^2 = 0.49\) and then multiply this by another \(0.7\) to get \((0.7)^3 = 0.343\). Continue multiplying by \(0.7\) to get \((0.7)^4 = 0.2401\) and finally \((0.7)^5 = 0.16807\).

Step 4: Multiply to find the charge

Next, multiply the result from Step 3 by 0.0003. Thus, \(C(5) = 0.0003 \times 0.16807 = 0.000050421\).

Step 5: Express the final result

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

Key Concepts

Function Evaluation

Function evaluation is like reading a recipe. You have the ingredients (the function and the value to substitute) and you follow the steps to find the outcome. In our exercise, we are given the function \(C(t) = 0.0003(0.7)^t\), which tells us how much charge is left in a battery after a certain number of days. By knowing the time span, say \(t = 5\), we perform function evaluation to find the exact charge left after these days.

Here's how you do it:

• Identify the value that you will substitute into the function; this is \(t = 5\) in this case.
• Carefully replace every \(t\) in your function with 5.
• Calculate step by step to discover the outcome.

Function evaluation requires accuracy and attention to detail. The ultimate goal is to arrive at a correct result that tells you about your specific interest—in this case, the battery charge after 5 days.

Exponentiation

Exponentiation sounds complex, but it’s simply the act of repeatedly multiplying a number by itself. In the given exercise, it demonstrates how the charge in a battery decreases over time. The function you are asked to evaluate entails raising 0.7 to the power of the number of days, \( t \).

Let’s break it down:

• When you see \((0.7)^t\), understand this means multiply 0.7 by itself \(t\) times.
• For \(t = 5\), you need to find \((0.7)^5\). Start with multiplying 0.7 by itself; do it step by step.
• First calculate \((0.7)^2 = 0.49\). Then, multiply by another 0.7 to get 0.343. Repeat in this manner to reach 0.16807 for our specific \(t\).

This process of exponentiation shows how something can reduce quickly or grow fast, depending on the base number. In the context of battery charge, the charge gets considerably smaller with each passing day.

Mathematical Modeling

Mathematical modeling helps us to simplify and represent real-world situations with equations. In this exercise, we used a function as a model to predict how the charge of a battery decays over time. This model is formulated as \(C(t) = 0.0003(0.7)^t\).

Why is this important?

• Models provide valuable insights into how systems behave under various conditions. They allow us to predict outcomes without direct measurement.
• The function \(C(t)\) abstracts the process where the battery loses its charge exponentially over days.
• Such models are used widely in various fields beyond chemistry and electronics; they can predict population changes, radioactive decay, and financial projections.

By using mathematical models, we can notably simplify complex systems, making it easier to make informed decisions based on the predicted data. In this case, understanding the battery decay model can guide users on how often they need to recharge their devices.