Problem 64 - Precalculus : Building Concepts and Connections

Question

Applications In this set of exercises, you will use properties of functions to study real-world problems. Depreciation The value of a computer \(t\) years after purchase is given by \(v(t)=2000-300 t,\) where \(v(t)\) is in dollars. Find the average rate of change of the value of the computer on the interval [0,3] , and interpret it.

Step-by-Step Solution

Answer

The average rate of change of the value of the computer over the first 3 years after purchase is \(-300\). That is, the value of the computer decreases by $300 per year on average.

Step 1: Understand the problem

The value of a computer \(t\) years after purchase is given by the equation \(v(t)=2000-300 t\), where \(v(t)\) is in dollars. The task is to determine the average rate of change of the value of the computer, specifically on the interval [0,3].

Step 2: Find the value of the function at given time points

The start and end time points on the interval [0,3] are \(t=0\) and \(t=3\) respectively. Firstly, substitute \(t=0\) into \(v(t)\) to get: \(v(0)=2000-300*0=2000\). Subsequently, substitute \(t=3\) into the function to get: \(v(3)=2000-300*3=1100\)

Step 3: Determine the average rate of change

The average rate of change is defined as the change in \(v(t)\) divided by the change in time \(t\) over a given interval. Mathematically, it can be calculated as: \[Average\: Rate\: of\: Change = \frac{v(3)-v(0)}{3-0}\]. Substituting the values calculated in Step 2, we get: \[Average\: Rate\: of\: Change = \frac{1100-2000}{3}\] which calculates to \(-300\).

Step 4: Interpret the result

The average rate of change is \(-300\). This means that, on average, the value of the computer decreases by $300 per year over the first 3 years after purchase.

Key Concepts

Depreciation

Depreciation describes how an asset, like a computer, loses value over time. When something depreciates, it means it becomes worth less. In our exercise, the formula given is \(v(t) = 2000 - 300t\).
This function explains how the computer’s value decreases each year.

• The initial value of the computer, when \(t=0\), is \\(2000\>.
• Each year, the computer loses \\)300 in value, indicated by the \(-300t\) part of the equation.

Understanding depreciation helps in planning finances. It shows when it might be wise to replace an item or reassess your budget.

Linear Function

A linear function is a simple mathematical expression where the output changes at a constant rate with respect to the input. In our problem, \(v(t) = 2000 - 300t\) is a linear function.

Key Properties

• Constant Rate of Change: Here, the value decreases by \$300 per year, consistent with a slope or rate of \(-300\).
• Graph Representation: The function would graph as a straight line, sloping downwards as time increases.
• Intercepts: The line crosses the y-axis at \(2000\), which is the starting value when \(t=0\).

Using linear functions in equations helps simplify complex problems by assuming a constant rate of change.

Real-World Application

Applying math concepts like depreciation and linear functions can solve everyday problems. This exercise measures how a computer’s value drops, useful for budgeting or deciding when to upgrade equipment.

Practical Uses

• Business Decisions: Companies can determine when to replace assets based on depreciation patterns.
• Financial Planning: Helps individuals predict the future value of their possessions.
• Economics: Understanding depreciation aids in assessing asset values for taxes and accounting.

By connecting math to real-life scenarios, students can see the relevance and usefulness of mathematics beyond the classroom.