Problem 91 - Intermediate Algebra

Question

The percent of the air-borne particles in a room that a Climatec furnace filter can remove is approximated by the linear function \(p(m)=\frac{6}{5} m\) where \(m\) is the time in minutes that the furnace has been operating. Use an inequality to determine the time after which at least \(60 \%\) of the air-borne particles in the room will have been removed. (Source: climatec.com)

Step-by-Step Solution

Answer

The furnace must operate for at least 50 minutes.

Step 1: Set Up the Inequality

We are given the function \( p(m) = \frac{6}{5}m \) which represents the percent of air-borne particles removed by the furnace filter after \( m \) minutes. To find the time after which at least 60% of the particles are removed, we need to set up the inequality \( \frac{6}{5}m \geq 60 \).

Step 2: Solve the Inequality

To find \( m \), solve the inequality \( \frac{6}{5}m \geq 60 \). Multiply both sides by \( \frac{5}{6} \) to isolate \( m \): \[ m \geq \frac{60 \times 5}{6} \].

Step 3: Perform the Calculation

Calculate the right side of the inequality: \[ m \geq \frac{60 \times 5}{6} = \frac{300}{6} = 50 \].

Step 4: Interpret the Result

The solution \( m \geq 50 \) means that the furnace needs to operate for at least 50 minutes to remove at least 60% of the air-borne particles.

Key Concepts

Linear functions

Linear functions are mathematical expressions used to describe situations where there is a constant rate of change. In our exercise, the function given is \(p(m) = \frac{6}{5}m\), which is a linear function representing the percentage of air-borne particles removed over time by a furnace filter. Here, \(m\) is the time in minutes, and \(\frac{6}{5}\) is the rate of change or slope.

• The function format for a linear equation is \(y = mx + c\). In this case, \(c = 0\) since our function starts at the origin (when \(m = 0\), \(p = 0\)).
• This implies a direct proportionality: as \(m\) increases by one, the percentage of particles removed increases by \(\frac{6}{5}\) percent.

Linear functions are easy to visualize as a straight line graph. Each point on this graph reflects a specific time duration and the corresponding percentage of particles filtered.

In practical terms, understanding how to interpret and use linear functions enables you to predict outcomes, such as determining how long the furnace must run to accomplish a desired filtration level.

Solving inequalities

Solving inequalities involves finding all possible values of a variable that satisfy the constraint given by the inequality. In our exercise, we were tasked to determine when the filtration function \(p(m)\) will exceed or equal 60%. This requires setting up and solving an inequality.

• The task is structured as \(\frac{6}{5}m \geq 60\).
• To solve, we first isolate \(m\) by multiplying both sides by the reciprocal of \(\frac{6}{5}\), which is \(\frac{5}{6}\).
• This transformation is crucial in simplifying the inequality to \(m \geq 50\).

After performing the calculation, it becomes evident that the furnace must run for at least 50 minutes to filter 60% of air-borne particles. This method of inequality solving is common in algebra and is particularly helpful in understanding constraints and limits in real-world problems.

Mastering inequalities allows you to understand when certain conditions are met, providing essential insights into planning and decision-making scenarios, just like determining how long a filter should operate.

Air-borne particles removal

Air-borne particles in the context of this exercise refer to tiny particles suspended in the air that can be harmful when inhaled. Removing these particles is a critical task for maintaining good indoor air quality. The effectiveness of devices like furnace filters is often quantified using mathematical models, such as the linear function discussed.

Key points in air filtration include:

• The total percentage of particles removed over time, which can be predicted using a linear function.
• Filters have varying efficiencies and time durations after which they reach certain filtration levels. This is where solving inequalities becomes valuable.

The real-world implication of achieving \(\geq60\%\) filtration is clear: by running the furnace for at least 50 minutes, air quality is noticeably improved.

Understanding these principles is essential for making informed decisions regarding air quality management—particularly in environments such as homes or offices where people spend a lot of time. It equips individuals and organizations to deploy air filtration systems effectively, enhancing health and comfort.