Ping Pong Ball Bounce Problem: Calculate The Initial Height

Hey guys! Let's dive into a fun math problem involving a bouncing ping pong ball. This problem is a classic example of a geometric sequence in action. We'll break it down step by step, making it super easy to understand. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into solving, let's make sure we fully grasp the situation. A ping pong ball is dropped from an unknown height. Each time it bounces, it only reaches one-third of the height of the previous bounce. After the third bounce, the ball reaches a height of 12 cm. Our mission, should we choose to accept it, is to figure out the initial height from which the ball was dropped. This involves understanding how the height decreases with each bounce, and then working backward to find the original height. Think of it like unraveling a mystery, where each bounce gives us a clue to the starting point.

Breaking Down the Bounces

To solve this, let's think about what happens with each bounce. If the ball bounces to one-third of the previous height, it means the height is being multiplied by 1/3 with each bounce. This is the key to understanding the geometric progression involved. Let's visualize this:

1. First Bounce: The ball reaches 1/3 of the initial height.
2. Second Bounce: The ball reaches 1/3 of the height it reached after the first bounce.
3. Third Bounce: The ball reaches 1/3 of the height it reached after the second bounce, and we know this height is 12 cm.

By understanding this pattern, we can reverse the process to find the initial height. This involves working backward from the third bounce, multiplying by the inverse of 1/3 (which is 3) to get the height of the previous bounce. This step-by-step approach is crucial for making the problem less daunting and more manageable. Each bounce is a step closer to solving the mystery of the initial height.

Why This Problem Matters

You might be thinking, "Why are we even doing this?" Well, besides being a fun brain teaser, this type of problem helps us understand real-world phenomena. Bouncing balls, decaying substances, and even compound interest follow similar patterns. By mastering this concept, we're not just solving a math problem; we're building a foundation for understanding various scientific and financial principles. Plus, it's a great way to sharpen our problem-solving skills, which are valuable in any field. So, let's keep going and unlock the solution together!

Setting Up the Equation

Alright, let's get down to the nitty-gritty and set up the equation. We know the ball bounces to 1/3 of its previous height each time. If we let 'h' represent the initial height, we can express the height after each bounce as follows:

• After the first bounce: h * (1/3)
• After the second bounce: h * (1/3) * (1/3) = h * (1/3)^2
• After the third bounce: h * (1/3) * (1/3) * (1/3) = h * (1/3)^3

We're given that the height after the third bounce is 12 cm. So, we can set up the equation:

h * (1/3)^3 = 12

This equation is the key to unlocking our answer. It represents the relationship between the initial height ('h') and the height after three bounces. By solving this equation, we'll find the value of 'h', which is the initial height we're looking for. Remember, the power of math lies in its ability to translate real-world scenarios into mathematical expressions that we can solve. This equation is a perfect example of that, taking the bouncing ball scenario and turning it into a solvable problem.

Why This Equation Works

You might be wondering why this equation is so effective. It all comes down to the properties of geometric sequences. Each bounce's height is a term in a geometric sequence where the common ratio is 1/3. The equation we've set up is essentially a formula for the nth term of a geometric sequence, but tailored to our specific problem. By using this formula, we can directly relate the final height (12 cm) to the initial height ('h'), making it much easier to solve the problem. It's like having a secret code that unlocks the answer! This connection between real-world scenarios and mathematical formulas is what makes math so powerful and applicable to various situations.

Preparing to Solve

Now that we have our equation, h * (1/3)^3 = 12, we're ready to start solving. But before we dive into the calculations, let's take a moment to think about the equation. We have 'h' multiplied by a fraction raised to a power, which equals 12. To isolate 'h', we'll need to undo these operations. This means we'll be using the concept of inverse operations, which is a fundamental tool in algebra. By understanding the structure of the equation and the operations involved, we can approach the solution with confidence and clarity. So, let's gear up and get ready to solve for 'h'!

Solving the Equation

Okay, let's tackle this equation step by step! We have h * (1/3)^3 = 12. First, let's simplify (1/3)^3. This means (1/3) * (1/3) * (1/3), which equals 1/27. So, our equation now looks like this:

h * (1/27) = 12

To isolate 'h', we need to get rid of the 1/27. Since 'h' is being multiplied by 1/27, we can do the opposite operation: multiply both sides of the equation by 27. This is a crucial step in solving algebraic equations – doing the same operation on both sides maintains the balance and keeps the equation true. When we multiply both sides by 27, we get:

h * (1/27) * 27 = 12 * 27

On the left side, (1/27) * 27 cancels out, leaving us with just 'h'. On the right side, 12 * 27 equals 324. So, we have:

h = 324

And there you have it! The initial height 'h' is 324 cm.

Breaking Down the Steps

Let's quickly recap the steps we took to solve the equation. First, we simplified the exponent (1/3)^3 to get 1/27. Then, we multiplied both sides of the equation by 27 to isolate 'h'. This systematic approach is key to solving algebraic equations effectively. By breaking down the problem into smaller, manageable steps, we can avoid confusion and ensure accuracy. Remember, each step has a purpose, and understanding the logic behind each step makes the process much smoother. So, let's celebrate our victory and move on to verifying our answer!

The Power of Inverse Operations

The technique we used to solve this equation highlights the power of inverse operations in algebra. Inverse operations are operations that undo each other. In this case, multiplication and division are inverse operations. By multiplying both sides of the equation by 27, we effectively undid the multiplication by 1/27, allowing us to isolate 'h'. This concept is fundamental to solving all sorts of algebraic equations, and mastering it will make you a much more confident problem solver. It's like having a secret weapon in your math arsenal!

Checking the Answer

Awesome! We've found that the initial height was 324 cm. But before we declare victory, let's double-check our answer. It's always a good idea to make sure our solution makes sense in the context of the problem. To do this, we can plug our answer back into the original problem and see if it holds true.

If the initial height is 324 cm, then after the first bounce, the ball would reach 1/3 of that height: 324 * (1/3) = 108 cm.

After the second bounce, it would reach 1/3 of 108 cm: 108 * (1/3) = 36 cm.

And after the third bounce, it would reach 1/3 of 36 cm: 36 * (1/3) = 12 cm.

This matches the information given in the problem – the ball reaches 12 cm after the third bounce. So, our answer of 324 cm is correct!

The Importance of Verification

Checking our answer is a crucial step in the problem-solving process. It's like the final brushstroke on a painting or the last note in a song. By verifying our solution, we ensure that we haven't made any mistakes along the way. This not only gives us confidence in our answer but also helps us develop a deeper understanding of the problem and the concepts involved. It's a habit that will serve you well in all areas of math and beyond. So, always remember to take that extra step and verify your solutions!

Different Ways to Verify

There are often multiple ways to verify your answer. In this case, we plugged our solution back into the original problem. Another way could be to use a different method to solve the problem and see if you get the same answer. The more ways you can verify your solution, the more confident you can be in its accuracy. It's like having multiple witnesses to confirm your findings. This flexibility in thinking and problem-solving is a valuable skill that will help you tackle even the most challenging problems.

Final Answer

Alright, guys! We've cracked the code! The initial height from which the ping pong ball was dropped is 324 cm. We solved the equation, verified our answer, and had some fun along the way. Math can be an adventure, and problems like these show us how math concepts connect to the real world.

Key Takeaways

Let's recap the key takeaways from this problem:

• Geometric Sequences: We used the concept of geometric sequences to understand how the height of the ball decreased with each bounce.
• Setting Up Equations: We translated the word problem into a mathematical equation that we could solve.
• Solving Equations: We used inverse operations to isolate the variable and find the solution.
• Verification: We checked our answer to ensure it made sense in the context of the problem.

These skills are valuable not only in math class but also in many real-life situations. By mastering these concepts, you'll become a more confident and capable problem solver.

Keep Exploring!

Math is like a vast and exciting landscape waiting to be explored. Don't stop here! There are so many more problems to solve and concepts to learn. Challenge yourself with similar problems, explore different areas of math, and never lose your curiosity. The more you explore, the more you'll discover the beauty and power of mathematics. Keep bouncing those ideas around, and who knows what amazing solutions you'll find!