Problem 143 - Algebra and Trigonometry

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Our hearts beat approximately 70 times per minute. Express in scientific notation how many times the heart beats over a lifetime of 80 years. Round the decimal factor in your scientific notation answer to two decimal places.

Step-by-Step Solution

Answer

The heart beats approximately \(2.94 * 10^{9}\) times over a lifespan of 80 years.

Step 1: Calculate total minutes in a lifetime

To calculate heart beats in a lifetime, we need to know total minutes in a lifetime. As there are 60 minutes in an hour, 24 hours in a day and 365 days in a year, multiply these values with the number of years in a lifetime: \(80 * 365 * 24 * 60\).

Step 2: Calculate total heartbeats in a lifetime

Knowing that the heart beats 70 times per minute, multiply the total minutes in a lifetime (calculated from step 1) with 70 to find the total heartbeats in a lifetime.

Step 3: Express the total heartbeats in scientific notation

Scientific notation is a method used to simplify large numbers. In scientific notation, a large number is converted into a decimal number between 1 and 10, then that decimal number is multiplied by a power of ten. Convert the result from step 2 into scientific notation. As per the exercise, round the decimal factor to 2 decimal places.

Key Concepts

Heartbeats Per Lifetime

To understand how many times your heart beats in a lifetime, we begin by considering the rate at which it beats. On average, a human heart beats about 70 times per minute. Over a lifetime—say an average of 80 years—this number adds up significantly. Calculating this can give fascinating insights into the body's continuous work.

First, calculate the total minutes in a lifetime. This requires knowledge of how many minutes are in a year. We know there are 60 minutes in every hour, 24 hours in each day, and 365 days in a year. Multiply these values by the 80 years of an average lifetime:

• 60 minutes/hour
• 24 hours/day
• 365 days/year

Multiplying all these figures together with the 80 years gives the total minutes over an entire lifetime. Then, multiply this by 70 (the average number of heartbeats per minute) to find the total heartbeats in 80 years.

Calculation of Large Numbers

Handling large numbers, such as the total beating of a heart over a lifetime, can become cumbersome and difficult to manage.

Once we multiply the total minutes in a lifetime by 70, we end up with a considerably large number. This is one reason why calculating such extensive numbers using step-by-step multiplication and conversion to simpler forms is helpful. Breaking down the calculations into smaller parts ensures accuracy and makes it easier to follow.

As outlined earlier:

• Calculate minutes for each year: 60 × 24 × 365
• Multiply by the total number of years (80)
• Multiply the result by 70 (heartbeats per minute)

These calculations lead us to a large figure representing the total heartbeats; an insightful exercise in managing significant numbers.

Scientific Notation Rounding

Scientific notation is a wonderful tool for simplifying the expression of very large or very small numbers.

When the total number of heartbeats is calculated, it results in a very large number. Expressing this in scientific notation involves converting that large number into a decimal between 1 and 10. This decimal is then multiplied by a power of ten.
Scientific notation allows for easier comprehension and comparison of vast numbers.
Moreover, rounding is necessary to present a cleaner, more manageable figure. Specifically, rounding the decimal part to two decimal places improves readability and precision without loss of significant information. This is crucial in scientific calculations where clarity and precision are paramount. Here, ensuring that the scientific notation is rounded properly means first converting, then rounding the decimal result to two decimal places as mentioned in the original exercise.

Minutes in a Lifetime

The concept of counting minutes to determine heartbeats is interrelated with our basic understanding of time.

To find out total minutes in a human lifespan, you need to consider several factors, including the number of minutes in an hour, hours in a day, days in a year, and potentially, years in a lifetime.

Here's how the calculation breaks down:

• 60 minutes in an hour
• 24 hours in a day
• 365 days in a year
• 80 years in a typical lifespan

When you multiply these numbers together, you get the total minutes present in 80 years. This foundational calculation is crucial for further computations, such as determining total heartbeats per lifetime or performing any long-term temporal analysis.