Well And Platform: Calculating Platform Height

Let's dive into a classic problem involving a well and a platform! This is a common type of question you might encounter, and it’s all about understanding volumes and how they relate to each other. So, grab your thinking caps, guys, and let’s break it down step by step.

Understanding the Problem

First, let's make sure we really get what's going on. We have a well, shaped like a cylinder, being dug into the ground. All the dirt that comes out of this well is then used to build a platform, which is essentially a rectangular prism (or a cuboid, if you want to get fancy). The key here is that the volume of the dirt taken out of the well must be equal to the volume of the platform that's built. This is because we're not adding or removing any dirt; we're just changing its shape.

The well has a diameter of 13 meters and a depth of 26 meters. This means the radius of the well is half the diameter, which is 6.5 meters. The platform has a length of 23 meters and a width of 14 meters. What we need to find is the height of this platform.

Key Information:

• Well:
  • Diameter = 13 meters
  • Radius = 6.5 meters
  • Depth (height) = 26 meters
• Platform:
  • Length = 23 meters
  • Width = 14 meters
  • Height = ? (This is what we need to find)

Calculating the Volume of the Well

The well is a cylinder, and the volume of a cylinder is given by the formula:

Volume = π * r^2 * h

Where:

• π (pi) is approximately 3.14159
• r is the radius of the base
• h is the height (or depth, in this case)

Let's plug in the values for the well:

Volume_well = π * (6.5)^2 * 26 Volume_well = π * 42.25 * 26 Volume_well = π * 1098.5 Volume_well ≈ 3450.29 cubic meters

So, the volume of the dirt taken out of the well is approximately 3450.29 cubic meters. Remember this number; it’s crucial! This step is important, guys, as this is where we find out how much soil we have to make the platform.

Calculating the Volume of the Platform

The platform is a rectangular prism (cuboid), and the volume of a rectangular prism is given by the formula:

Volume = l * w * h

Where:

• l is the length
• w is the width
• h is the height

We know the length and width of the platform, but we don't know the height. That's what we're trying to find! Let's plug in the values we know:

Volume_platform = 23 * 14 * h Volume_platform = 322 * h

Equating the Volumes and Solving for the Height

Here's the key step: The volume of the dirt from the well must equal the volume of the platform. So, we can set the two volume equations equal to each other:

Volume_well = Volume_platform 3450.29 = 322 * h

Now, we can solve for h (the height of the platform) by dividing both sides of the equation by 322:

h = 3450.29 / 322 h ≈ 10.72 meters

Therefore, the height of the platform is approximately 10.72 meters. And that's our answer! This is the final calculation, guys. You have solved the problem.

Checking Our Answer

It's always a good idea to double-check our work, just to be sure. Let's plug the height we found back into the volume equation for the platform:

Volume_platform = 23 * 14 * 10.72 Volume_platform = 322 * 10.72 Volume_platform ≈ 3452.44 cubic meters

This is very close to the volume of the well (3450.29 cubic meters). The slight difference is due to rounding errors during our calculations. But overall, it looks like our answer is correct! Always double-check, guys. It really makes the difference.

Key Takeaways

• Volume is conserved: When you move material from one place to another and reshape it, the volume stays the same (assuming you don't add or remove any material).
• Formulas are your friends: Knowing the formulas for the volumes of common shapes (cylinder, rectangular prism, etc.) is essential for solving these types of problems.
• Units are important: Make sure you're using the same units throughout the problem (meters in this case). If you mix units, you'll get the wrong answer.
• Double-check your work: It's always a good idea to double-check your calculations to avoid errors.
• Understand the problem: Reading carefully and understanding the context is more important than immediately trying to find a solution.

Common Mistakes to Avoid

• Using the diameter instead of the radius: Remember that the radius is half the diameter. Using the diameter in the volume formula will give you the wrong answer.
• Forgetting the units: Always include the units in your answer (meters in this case). This helps you avoid mistakes and makes your answer more clear.
• Rounding errors: Rounding too early in the calculation can lead to significant errors in the final answer. Try to keep as many decimal places as possible until the very end.
• Incorrect formulas: Make sure you're using the correct formulas for the volumes of the shapes involved. A wrong formula will, obviously, lead to a wrong answer.

Real-World Applications

This type of problem might seem abstract, but it has real-world applications in construction, engineering, and other fields. For example, engineers might need to calculate the amount of dirt needed to build a foundation for a building, or the amount of concrete needed to pour a sidewalk. It’s all connected, guys! Understanding how to calculate volumes is a valuable skill.

Practice Problems

Want to test your understanding? Try these practice problems:

1. A cylindrical tank with a diameter of 8 meters and a height of 5 meters is filled with water. The water is then poured into a rectangular tank with a length of 10 meters and a width of 4 meters. What is the height of the water in the rectangular tank?
2. A conical pile of sand has a radius of 3 meters and a height of 2 meters. The sand is then used to create a level surface with an area of 20 square meters. What is the thickness of the sand layer?

Conclusion

So, there you have it! Calculating the height of a platform made from the dirt of a well is a matter of understanding volume, applying the correct formulas, and paying attention to detail. With a little practice, you'll be solving these problems like a pro! Keep practicing, guys, and you'll get there! These type of problems are all about practice. The more you repeat the steps, the easier it becomes to get the correct answer. Good luck!