Problem 76 - Algebra A Combined Function

Question

The human eye blinks once every 5 seconds on average. How many times does the average eye blink in one hour? In one 16 -hour day while awake? In one year while awake?

Step-by-Step Solution

Answer

The average eye blinks 720 times per hour, 11,520 times in a 16-hour day, and 4,204,800 times in a year while awake.

Step 1: Calculate Blinks Per Minute

The average eye blinks once every 5 seconds. Since there are 60 seconds in a minute, the number of blinks in a minute is calculated by dividing 60 by 5.\[\frac{60}{5} = 12\]Thus, the average eye blinks 12 times per minute.

Step 2: Calculate Blinks Per Hour

Knowing that the average eye blinks 12 times per minute, we need to determine the number of blinks per hour. Since there are 60 minutes in an hour, we multiply the blinks per minute by the number of minutes in an hour.\[12 \text{ blinks per min} \times 60 \text{ min} = 720 \text{ blinks}\]So, the average eye blinks 720 times in an hour.

Step 3: Calculate Blinks in a 16-hour Day

Given that the average person is awake for about 16 hours, we calculate the total number of blinks in one day by multiplying the blinks per hour by the number of waking hours.\[720 \text{ blinks per hour} \times 16 \text{ hours} = 11,520 \text{ blinks}\]Therefore, the average eye blinks 11,520 times in a 16-hour day.

Step 4: Calculate Blinks in a Year

First, we calculate how many days are considered waking days in a year. Assuming 365 days in a year:\[11,520 \text{ blinks per day} \times 365 \text{ days} = 4,204,800 \text{ blinks}\]Thus, the average eye blinks around 4,204,800 times in a year while awake.

Key Concepts

Rate Calculations

Rate calculations are essential for understanding how frequently an event occurs over a particular period. In this context, we are calculating the rate of eye blinks per unit of time. When we say an eye blinks once every 5 seconds, we are describing a rate. To convert this to blinks per minute, you simply divide the total seconds in a minute by the seconds per blink. In our example, we calculated

• 60 seconds in a minute divided by 5 seconds per blink = 12 blinks per minute.

This calculation helps us establish a foundational rate that we can use to further determine blinks in larger time frames like an hour or a day. Let's remember:

• To find blinks per hour, multiply blinks per minute by 60 (minutes in an hour).
• To find blinks in any period, always use the base rate (here, 12 blinks per minute) for extrapolation.

Mastering rate calculations allows you to analyze repetitive events systematically.

Unit Conversion

Unit conversion is an important part of solving algebra word problems, especially those involving rates. Here, proper conversions allow for precise calculations when shifting from one unit of time to another. We start by converting blinks per second to blinks per minute, and then to blinks per hour.

Key conversion steps include:

• Seconds to minutes: Since there are 60 seconds in a minute, divide or multiply as necessary to switch units.
• Minutes to hours: There are 60 minutes in one hour, so use multiplication to scale your findings accordingly.

It's crucial to accurately perform these conversions to ensure your calculations remain consistent with the units being converted. This skill is not just isolated to eye blinks but is applicable to any scenario requiring multiple time unit calculations.

Multiplication in Algebra

Multiplication is a fundamental concept in algebra used to expand basic calculations across larger quantities or more complex formulas. In word problems like our example, multiplication allows for the scaling of rates to longer periods.

• First, we found the rate: 12 blinks per minute. We then used multiplication to project this rate over greater timeframes, like 60 minutes (an hour) or 16 hours (a waking day).
• Our other multiplication involved day to year conversion, taking the daily blink total and scaling it for 365 days.

Effective use of multiplication in algebra not only provides accurate results but also simplifies repetitive addition, making calculations cleaner and more efficient. Practicing with real-world examples like the blink rate helps solidify these skills in a practical context.