Problem 19 - Algebra Form and Function

Question

The value of an antique is \(2500+80 n\) dollars, where \(n\) is the number of years since the antique is purchased. A professor calculates a homework grade of \(100-3 n\) for \(n\) missing homework assignments.

Step-by-Step Solution

Answer

Answer: To find the value of an antique after 7 years, we will use the formula: Value of antique = 2500 + 80n Substitute the value of n with the number of years, which is 7: Value of antique = 2500 + 80(7) Now, calculate the value of the antique: Value of antique = 2500 + 560 = 3060 So, after 7 years, the value of the antique will be $3060. To calculate the homework grade for a student who missed 4 assignments, we will use the formula: Homework grade = 100 - 3n Substitute the value of n with the number of missing assignments, which is 4: Homework grade = 100 - 3(4) Now, calculate the homework grade: Homework grade = 100 - 12 = 88 So, the homework grade for this student with 4 missing assignments will be 88.

Step 1: Problem 1: Calculate the value of the antique after a certain number of years

To find the value of the antique after a specific number of years, we will use the formula: Value of antique = \(2500 + 80n\) Where \(n\) is the number of years since the antique was purchased.

Step 1: Substitute the value of n

For a given number of years, plug the value of \(n\) into the formula. For example, let's calculate the value of the antique after 5 years. Value of antique = \(2500 + 80(5)\)

Step 2: Calculate the value of the antique

Now, calculate the value of the antique by solving the equation: Value of antique = \(2500 + 80(5) = 2500 + 400 = 2900\) So, after 5 years, the value of the antique will be $2900.

Step 4: Problem 2: Calculate the homework grade of a student based on the number of missing homework assignments

To find the homework grade based on the number of missing homework assignments, we will use the formula: Homework grade = \(100 - 3n\) Where \(n\) represents the number of missing homework assignments.

Step 1: Substitute the value of n

For a given number of missing homework assignments, plug the value of \(n\) into the formula. For example, let's calculate the homework grade for a student who missed 3 assignments. Homework grade = \(100 - 3(3)\)

Step 2: Calculate the homework grade

Now, calculate the homework grade by solving the equation: Homework grade = \(100 - 3(3) = 100 - 9 = 91\) So, the homework grade for this student with 3 missing assignments will be \(91\).

Key Concepts

Value Calculation

Understanding how to calculate the value of an asset over time is an essential skill in everyday situations like investments, collecting antiques, or even assessing how items depreciate. In the given exercise, the value of an antique increases over time using a linear equation. The equation given is \( V = 2500 + 80n \), where \( V \) is the value in dollars and \( n \) is the number of years since purchase.

This equation represents a linear growth, meaning each year adds a constant amount to the initial value. Let's break down the process:

• Start by identifying the known values: Here, always begin with the initial value, which is \( 2500 \).
• Understand the annual growth: This example shows the antique's value increasing by \( 80 \) dollars per year.

To find the value after a set number of years, simply multiply the annual growth by the number of years since purchase and add this to the initial value. For example, after 5 years, you would calculate \( 2500 + 80 \times 5 = 2900 \), meaning the antique's value is $2900 then.

Homework Grading

The concept of grading homework based on the number of assignments missed can be illustrated using another simple linear equation. This provides students a clear view of how grades can be systematically affected by missing work. The formula for the homework grade is \( G = 100 - 3n \), where \( G \) represents the grade, and \( n \) is the number of assignments missed.

Let's see how this works:

• Start with a perfect score: The base grade before deductions is always 100.
• Deductions per missed assignment: In this formula, 3 points are deducted for each missed homework assignment.

To calculate the grade after missing assignments, you substitute \( n \) with the number of missed assignments. For example, if a student missed 3 assignments, plug it into the equation: \( 100 - 3 \times 3 = 91 \). The student's final grade would be 91. This clear method helps emphasize accountability in completing assignments.

Substitution Method

The substitution method is a viable strategy for solving equations when you want to find specific values in scenarios like calculating antique values or determining homework grades. This straightforward technique involves replacing variables in equations with known or chosen numbers to swiftly find a solution.

Here's how the substitution method works:

• Identify the equation and the variable you need to calculate.
• Choose the known variable's value to substitute into this equation.

For instance, in the equation for the antique's value \( 2500 + 80n \), if you're tasked with finding the value after 5 years, substitute \( n = 5 \) into the equation, making it \( 2500 + 80 imes 5 \). The same principle applies to the homework grading formula \( 100 - 3n \). If a student has missed 3 assignments, substitute \( n = 3 \) to find the grade.

Utilizing substitution, you convert an abstract problem into a specific calculation, simplifying and solving the problem step by step.