Calculating Total Weight The Approximate Weights Of Two Animals In Scientific Notation

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Hey guys! Today, let's dive into a fascinating math problem that involves calculating the total weight of two super heavy animals using scientific notation. This is a common type of problem in science and engineering, and mastering it will definitely boost your math skills. So, let's get started!

Understanding the Problem

The problem states that we have two animals with approximate weights of 5.36 x 10^4 lbs and 6.2 x 10^4 lbs. Our mission is to find the total weight of these two magnificent creatures and express our final answer in scientific notation, while also making sure we use the correct number of significant digits. Sounds exciting, right?

What is Scientific Notation?

Before we jump into the calculations, let's quickly recap what scientific notation is all about. Scientific notation is a neat way of expressing very large or very small numbers in a compact and standardized form. It's like a mathematical shorthand! The general form of scientific notation is:

a x 10^b

Where:

  • a is a number between 1 and 10 (but not including 10). This is called the coefficient or significand.
  • 10 is the base.
  • b is an integer (positive or negative), known as the exponent or power of 10.

For instance, the number 53,600 can be written in scientific notation as 5.36 x 10^4. See how we've expressed a large number in a more manageable way? This is super useful when we're dealing with astronomical figures or microscopic measurements!

Significant Digits: Why They Matter

Now, let's talk about significant digits. Significant digits are the digits in a number that carry meaningful information about its precision. They tell us how accurately a measurement or calculation has been made. When we're working with scientific notation, it's crucial to maintain the correct number of significant digits in our final answer.

Here are a few key rules to remember about significant digits:

  1. All non-zero digits are significant. For example, in the number 3.14159, all six digits are significant.
  2. Zeros between non-zero digits are significant. For instance, in the number 4007, all four digits are significant.
  3. Leading zeros (zeros to the left of the first non-zero digit) are not significant. For example, in the number 0.0052, only the 5 and 2 are significant.
  4. Trailing zeros (zeros to the right of the last non-zero digit) are significant if the number contains a decimal point. For example, in the number 12.300, all five digits are significant. If there's no decimal point, trailing zeros might or might not be significant, depending on the context. Usually, we assume they are not significant unless indicated otherwise.

Why do significant digits matter? Well, they ensure that our calculations reflect the accuracy of our initial measurements. If we start with numbers that have a certain level of precision, we don't want our final answer to suggest a higher level of precision than we actually have.

Solving the Problem: Step-by-Step

Alright, let's get our hands dirty and solve this problem! Here’s a step-by-step breakdown:

Step 1: Adding the Weights

Our first task is to add the weights of the two animals. Since both weights are already expressed in scientific notation with the same power of 10 (10^4), we can directly add the coefficients:

5.36 x 10^4 lbs + 6.2 x 10^4 lbs

To add these, we simply add the numbers in front (the coefficients) and keep the 10^4 part the same:

(5.36 + 6.2) x 10^4 lbs

Now, let's do the addition:

5. 36 + 6.2 = 11.56

So, we have:

11. 56 x 10^4 lbs

Step 2: Adjusting to Scientific Notation

Great! We've added the weights, but our answer isn't quite in proper scientific notation yet. Remember, in scientific notation, the coefficient needs to be a number between 1 and 10. Currently, our coefficient is 11.56, which is larger than 10. So, we need to make an adjustment.

To do this, we'll move the decimal point one place to the left in the coefficient. This makes our coefficient smaller, but to keep the overall value the same, we need to increase the exponent by 1. Here's how it looks:

11. 56 x 10^4 lbs = 1.156 x 10^(4+1) lbs

= 1.156 x 10^5 lbs

Now, our number is in proper scientific notation!

Step 3: Significant Digits: The Final Touch

We're almost there! The last step is to make sure our answer has the correct number of significant digits. Looking back at our original numbers:

  • 5.36 x 10^4 lbs has three significant digits (5, 3, and 6).
  • 6.2 x 10^4 lbs has two significant digits (6 and 2).

When we add or subtract numbers, the result should have the same number of decimal places as the number with the fewest decimal places. However, in this case, we need to consider the number of significant digits in the original numbers. The number with the fewest significant digits is 6.2 x 10^4, which has two significant digits.

So, our final answer should also have two significant digits. We need to round 1.156 x 10^5 lbs to two significant digits. To do this, we look at the third digit (5). Since it's 5 or greater, we round up the second digit:

1. 156 x 10^5 lbs ≈ 1.2 x 10^5 lbs

The Final Answer

And there we have it! The total weight of the two animals, expressed in scientific notation with the correct number of significant digits, is:

1. 2 x 10^5 lbs

Why This Matters: Real-World Applications

Guys, you might be wondering,