Problem 101 - College Algebra and Calculus: An Applied Approach

Question

The specifications for an electronic device state that it is to be operated in a room with relative humidity \(h\) defined by \(|h-50| \leq 30 .\) What are the minimum and maximum relative humidities for the operation of this device?

Step-by-Step Solution

Answer

The minimum operational relative humidity is 20% and the maximum is 80%

Step 1: Understand the Absolute Value Inequality

The inequality \(|h - 50| \leq 30\) means the difference between \(h\) (relative humidity) and 50 is no more than 30. This results in two possible scenarios, either:\(h - 50 \leq 30\) or \(-30 \leq h - 50\). Now, simplify these two inequalities to find the range of relative humidity.

Step 2: Solve the Equality \(h - 50 \leq 30\)

To find the maximum relative humidity, solve \(h - 50 \leq 30\). Add 50 on both sides of the inequality to isolate \(h\). The resulting inequality \(h \leq 80\) is the maximum range of the relative humidity.

Step 3: Solve the Equality \(-30 \leq h - 50\)

To find the minimum relative humidity, solve \(-30 \leq h - 50\). Again, add 50 to both sides of the inequality to isolate \(h\). The resulting inequality \(h \geq 20\) is the minimum range of the relative humidity.

Step 4: Conclude the Result

The relative humidity \(h\) must be greater than or equal to 20 and at the same time be less than or equal to 80. So, the minimum and maximum relative humidities for the operation of the device are 20% and 80% respectively.

Key Concepts

Relative Humidity

Relative humidity is a measure of how much moisture the air contains compared to the maximum amount it can hold at a specific temperature. It's a crucial factor for many electronic devices as it can affect their performance and longevity.
Think of it like a sponge soaking up water:

• If the sponge is almost full, it's like high humidity.
• If it's nearly empty, it's like low humidity.

For electronic devices, controlling the humidity level ensures they operate smoothly without damage from either too much moisture or too dry conditions.

Solving Inequalities

Solving inequalities involves finding the range of values that satisfy a given condition. In the context of absolute value inequalities, like \(|h - 50| \leq 30\), this means finding both the upper and lower limits for the value inside the absolute value.
There are a couple of key steps:

• Understand that \(|h - 50| \leq 30\) splits into \(h - 50 \leq 30\) and \(-30 \leq h - 50\).
• Solve each inequality separately to find the limits of \(h\).
• This approach helps pinpoint the exact range \(h\) can be, ensuring the device operates optimally.

Taking it step by step allows you to capture all possible solutions effectively.

Range of Values

The range of values refers to the set of possible solutions that satisfy an inequality or equation. For the device, it's critical to maintain the humidity level between 20% and 80% to ensure reliable operation.
Let's break it down:

• Solve \(h - 50 \leq 30\), which simplifies to \(h \leq 80\), giving the upper limit.
• Solve \(-30 \leq h - 50\), which simplifies to \(h \geq 20\), providing the lower limit.

The combination of these results gives us \(20 \leq h \leq 80\). This range ensures the device stays within optimal humidity levels, preventing damage and maintaining efficiency.