Problem 75 - Algebra A Combined Function

Question

Only male crickets chirp. They chirp at different rates depending on their species and the temperature of their environment. Suppose a certain species is currently chirping at a rate of 90 chirps per minute. At this rate, how many chirps occur in one hour? In one 24 -hour day? In one year?

Step-by-Step Solution

Answer

5400 chirps per hour, 129600 chirps per day, and 47,304,000 chirps per year.

Step 1: Calculate Chirps per Hour

Since the crickets chirp at a rate of 90 chirps per minute, to find the number of chirps per hour, multiply the number of chirps per minute by the number of minutes in an hour: \[ 90 \, \text{chirps/minute} \times 60 \, \text{minutes} = 5400 \, \text{chirps/hour} \]

Step 2: Calculate Chirps per Day

To find the number of chirps in a 24-hour day, multiply the number of chirps per hour by the number of hours in a day: \[ 5400 \, \text{chirps/hour} \times 24 \, \text{hours} = 129600 \, \text{chirps/day} \]

Step 3: Calculate Chirps per Year

To calculate the number of chirps in a year, multiply the number of chirps per day by the number of days in a year (using 365 days for simplicity): \[ 129600 \, \text{chirps/day} \times 365 \, \text{days} = 47,304,000 \, \text{chirps/year} \]

Key Concepts

Multiplication

Multiplication is a fundamental arithmetic operation that allows us to calculate the result of adding a number to itself a certain number of times. In this exercise, we need to find out how many times crickets chirp over different time periods, starting with per hour. The cricket chirp rate is given as 90 chirps per minute. So, we multiply 90 chirps by the number of minutes in an hour, which is 60, to determine the total chirps in one hour.

Mathematically, it's expressed as:

• Number of chirps per hour = 90 chirps/minute × 60 minutes
• Using multiplication helps us quickly compute this as 5400 chirps/hour.

This method of repeated addition, through multiplication ensures quick calculation of large totals without physically adding repeatedly, making it a valuable mathematical tool.

Unit Conversion

Unit conversion is essential when dealing with different measurements across a problem. In our exercise, the initial chirp rate is given per minute. Converting this to other units, like per hour, day, or year, requires understanding and manipulating time units.

The process involves:

• Converting minutes to hours, by knowing there are 60 minutes in one hour.
• Scaling the hourly rate up to a daily figure, using the fact that there are 24 hours in a day.
• Finally, extending the daily chirp rate to cover a year, assuming a year equals 365 days.

Efficiently navigating these conversions ensures accuracy in contexts such as scientific analyses or real-world applications like understanding animal behaviors over time. Unit conversion streamlines these tasks by enabling seamless transitions between different units without altering the fundamental rate or measure of the activity observed.

Word Problems

Word problems like this one challenge us to apply mathematical concepts in real-life scenarios. They require understanding the context, extracting relevant numerical information, translating that into mathematical operations, and performing calculations methodically.

Key steps include:

• Identifying the rate of chirping given as 'per minute'.
• Determining the needed time scale changes: hourly, daily, and yearly.
• Accurately performing calculations to adapt to these scales.

Through this structured approach, word problems test our ability to inquire and solve. However, they are a great method to develop critical thinking skills by engaging with numbers in everyday circumstances. Understanding word problems encourages analyzing similarities and differences in scenario-driven mathematics, making us better problem solvers.