Solving For Time How Long To Dig A Well Equation Explained
Hey everyone! Today, we're diving into a fun math problem that involves digging a well. Imagine Pieter, a hardworking fellow, who's trying to figure out how long it'll take to dig a well 72 feet below sea level. He's written down an equation, 7h - 5(3h - 8) = -72, and our mission is to help him solve it. So, grab your thinking caps, and let's get started!
Understanding the Equation
First, let's break down the equation: 7h - 5(3h - 8) = -72. This equation represents a real-world scenario, and understanding its parts is crucial. The variable 'h' stands for the number of hours it will take to dig the well. The equation itself shows the relationship between the hours worked and the depth of the well. The left side of the equation, 7h - 5(3h - 8), represents the total work done, considering both the time spent digging and any potential setbacks or breaks. The right side, -72, indicates the final depth of the well below sea level (the negative sign signifies 'below').
When you first look at this equation, it might seem a bit daunting, but don't worry! We're going to tackle it step by step. The key here is to remember the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order tells us the sequence in which we need to perform the calculations to solve the equation correctly. In our case, we'll start by simplifying the expression within the parentheses, then handle the multiplication, and finally, combine like terms to isolate the variable 'h'. This methodical approach will make the process much clearer and less intimidating.
Step-by-Step Solution
Now, let's walk through the solution together. Remember, the goal is to isolate 'h' on one side of the equation to find its value. Here's how we'll do it:
1. Distribute the -5
The first step in solving this equation is to simplify the expression by distributing the -5 across the terms inside the parentheses. This means we'll multiply -5 by both 3h and -8. Doing so, we get:
7h - 5(3h - 8) = -72
7h - 15h + 40 = -72
Notice how multiplying -5 by -8 gives us a positive 40. It's essential to pay attention to the signs when performing these operations. Distributing correctly is a crucial step as it clears the way for combining like terms and further simplifying the equation.
2. Combine Like Terms
Next, we need to combine the like terms on the left side of the equation. In this case, we have two terms with 'h': 7h and -15h. Combining these gives us:
7h - 15h + 40 = -72
-8h + 40 = -72
Combining like terms simplifies the equation, making it easier to isolate the variable. By reducing multiple terms into a single term, we're one step closer to solving for 'h'. This step is all about tidying up the equation and making it more manageable.
3. Isolate the Variable Term
Our next task is to isolate the term containing 'h'. Currently, we have -8h + 40 = -72. To get -8h by itself, we need to get rid of the +40. We can do this by subtracting 40 from both sides of the equation:
-8h + 40 - 40 = -72 - 40
-8h = -112
Remember, whatever we do to one side of the equation, we must do to the other to maintain the equality. Subtracting 40 from both sides ensures that the equation remains balanced while moving us closer to isolating 'h'.
4. Solve for h
Finally, to solve for 'h', we need to get it completely by itself. We have -8h = -112. Since 'h' is being multiplied by -8, we'll do the opposite operation and divide both sides of the equation by -8:
-8h / -8 = -112 / -8
h = 14
Dividing both sides by -8 isolates 'h' and gives us the solution. Remember that a negative divided by a negative results in a positive, so -112 divided by -8 equals 14. This means it will take Pieter 14 hours to dig the well to a level of 72 feet below sea level.
The Answer
So, after all that math, we've found that h = 14. This means it will take Pieter 14 hours to dig the well. Looking back at our multiple-choice options, the correct answer is C. 14. Great job, guys! We've successfully navigated through the equation and found the solution. This problem demonstrates how algebraic equations can model real-world situations, and by following a step-by-step approach, we can solve even the trickiest problems.
Real-World Application
This problem isn't just about abstract math; it's about applying mathematical concepts to real-world scenarios. Think about it: Pieter needed to figure out the time required for a specific task, and he used an equation to model the situation. This is something we do all the time in various fields, from engineering and construction to finance and economics. Understanding how to set up and solve equations is a valuable skill that can help us make informed decisions and solve practical problems.
Imagine you're planning a construction project. You need to estimate the time it will take to complete various tasks, the resources required, and the overall cost. Equations like the one we solved today can be used to model these relationships and make accurate predictions. For example, you might have an equation that relates the number of workers, the amount of materials, and the time it takes to build a structure. By plugging in different values, you can explore various scenarios and optimize your plan.
Moreover, understanding these concepts is crucial in everyday life. Whether you're calculating the time it takes to drive to a destination, figuring out the cost of a shopping trip, or even estimating the cooking time for a recipe, math is involved. By strengthening your algebra skills, you're not just acing math tests; you're equipping yourself with tools to navigate the world more effectively. So, keep practicing, keep exploring, and keep applying math to real-life situations. You'll be amazed at how powerful these skills can be!
Why This Matters
Understanding how to solve equations like this one is more than just an academic exercise. It's a fundamental skill that empowers you to tackle real-world problems. Think about it: equations are mathematical models of reality. They help us understand relationships between different quantities and make predictions. Whether you're calculating how long it will take to drive to a destination, figuring out the cost of a project, or even determining the optimal dose of medicine, equations are at the heart of the process.
The beauty of algebra, and mathematics in general, is its ability to abstract complex situations into simpler, manageable forms. An equation like 7h - 5(3h - 8) = -72 might seem intimidating at first glance, but by breaking it down into steps, we can unravel its meaning and find a solution. This process of simplification and problem-solving is a skill that translates far beyond the classroom. It's about critical thinking, logical reasoning, and the ability to approach challenges with a structured mindset. These are qualities that are highly valued in any field, from science and technology to business and the arts.
Furthermore, the ability to solve equations builds confidence. When you successfully tackle a challenging problem, you develop a sense of accomplishment and a belief in your own abilities. This confidence can spill over into other areas of your life, making you more willing to take on new challenges and persevere through difficulties. So, the next time you encounter an equation, remember that it's not just a set of symbols; it's a puzzle waiting to be solved, a challenge waiting to be conquered, and an opportunity to grow your skills and confidence.
Practice Makes Perfect
Like any skill, solving equations gets easier with practice. The more you work through different types of problems, the more comfortable you'll become with the process. Don't be afraid to make mistakes – they're a natural part of learning. Each mistake is an opportunity to understand where you went wrong and how to improve. There are tons of resources available to help you practice, from textbooks and online tutorials to interactive games and apps. Find what works best for you and make practice a regular part of your routine.
Consider working through similar problems with slight variations. For instance, you might change the numbers in the equation or adjust the context of the problem. What if the well was being dug on a different planet with varying gravitational forces? How would that affect the equation? What if there were unexpected delays or equipment malfunctions? By exploring these scenarios, you'll deepen your understanding of the underlying concepts and develop your problem-solving skills even further. Remember, the goal isn't just to memorize steps but to truly understand the math behind them. The more you practice, the more intuitive these concepts will become, and the more confident you'll feel when faced with new challenges.
So, there you have it! We've successfully solved Pieter's well-digging equation and found that it will take him 14 hours to dig the well. Remember, the key is to break down the problem into smaller, manageable steps and to apply the order of operations correctly. Keep practicing, and you'll become a master equation solver in no time! If you have any questions or want to explore more math problems, feel free to ask. Happy problem-solving!
