Calculating Water Depth In A Rectangular Container A Step By Step Guide

In this article, we will explore a practical problem involving the transfer of water between containers of different shapes. Specifically, we will calculate the depth of water in a rectangular container after it has been poured from a cylindrical container. This problem combines concepts of geometry, volume calculation, and unit conversion, making it a valuable exercise for understanding these principles. Let's dive into the details.

Problem Statement

A cylindrical container has a height of 28 cm and a diameter of 18 cm and is completely filled with water. The water from this cylindrical container is then poured into another container that has a rectangular base. The rectangular container has a length of 27 cm and a width of 11 cm. Our task is to calculate the depth of the water in the rectangular container after the transfer.

Understanding the Concepts

Before we proceed with the calculations, let's ensure we understand the key concepts involved in this problem:

1. Volume of a Cylinder: The volume of a cylinder is the amount of space it occupies. It is calculated using the formula:

   $$V_{cylinder} = \pi r^2 h$$

   where:

   • $V_{cylinder}$ is the volume of the cylinder,
   • $\pi$ (pi) is a mathematical constant approximately equal to 3.14159,
   • $r$ is the radius of the cylinder's base, and
   • $h$ is the height of the cylinder.

2. Volume of a Rectangular Prism: The volume of a rectangular prism (which is the shape of the rectangular container) is the amount of space it occupies. It is calculated using the formula:

   $$V_{rectangular} = l \times w \times d$$

   where:

   • $V_{rectangular}$ is the volume of the rectangular prism,
   • $l$ is the length of the rectangular base,
   • $w$ is the width of the rectangular base, and
   • $d$ is the depth (or height) of the water in the rectangular container.

3. Conservation of Volume: When water is poured from one container to another, the volume of the water remains the same. This principle is crucial for solving this problem. The volume of water in the cylindrical container will be equal to the volume of water in the rectangular container.

Step-by-Step Solution

Now that we have a clear understanding of the concepts, let's solve the problem step by step:

Step 1: Calculate the Volume of the Cylindrical Container

First, we need to find the volume of water in the cylindrical container. We are given the height ($h$) as 28 cm and the diameter as 18 cm. To use the volume formula, we need the radius ($r$), which is half of the diameter.

• Radius, $r = \frac{diameter}{2} = \frac{18 \text{ cm}}{2} = 9 \text{ cm}$

Now we can calculate the volume of the cylinder using the formula:

$$V_{cylinder} = \pi r^2 h$$

Substituting the given values:

$$V_{cylinder} = \pi (9 \text{ cm})^2 (28 \text{ cm})$$

$$V_{cylinder} = \pi (81 \text{ cm}^2) (28 \text{ cm})$$

$$V_{cylinder} = 2268\pi \text{ cm}^3$$

Using the approximation $\pi \approx 3.14159$, we get:

$$V_{cylinder} \approx 2268 \times 3.14159 \text{ cm}^3$$

$$V_{cylinder} \approx 7129.38 \text{ cm}^3$$

So, the volume of water in the cylindrical container is approximately 7129.38 cubic centimeters.

Step 2: Calculate the Depth of Water in the Rectangular Container

Next, we need to find the depth of the water in the rectangular container. We know the length ($l$) of the rectangular base is 27 cm, the width ($w$) is 11 cm, and the volume of water in the rectangular container ($V_{rectangular}$) is the same as the volume of water from the cylinder (since volume is conserved). Therefore,

$$V_{rectangular} = V_{cylinder} \approx 7129.38 \text{ cm}^3$$

The formula for the volume of a rectangular prism is:

$$V_{rectangular} = l \times w \times d$$

We need to find the depth ($d$). Rearranging the formula to solve for $d$:

$$d = \frac{V_{rectangular}}{l \times w}$$

Substituting the known values:

$$d = \frac{7129.38 \text{ cm}^3}{27 \text{ cm} \times 11 \text{ cm}}$$

$$d = \frac{7129.38 \text{ cm}^3}{297 \text{ cm}^2}$$

$$d \approx 24.00 \text{ cm}$$

Therefore, the depth of the water in the rectangular container is approximately 24.00 cm.

Verification

To verify our answer, we can recalculate the volume of the rectangular container using the calculated depth and check if it matches the volume of the cylindrical container.

$$V_{rectangular} = l \times w \times d$$

$$V_{rectangular} = 27 \text{ cm} \times 11 \text{ cm} \times 24.00 \text{ cm}$$

$$V_{rectangular} = 7128 \text{ cm}^3$$

This volume is very close to the volume of the cylindrical container (7129.38 cm³), which confirms our calculation is accurate.

Conclusion

In this article, we successfully calculated the depth of water in a rectangular container after it was poured from a cylindrical container. We found that the depth of the water in the rectangular container is approximately 24.00 cm. This problem demonstrated the application of geometric formulas for volume calculation and the principle of volume conservation. Understanding these concepts is essential for solving various real-world problems related to fluid mechanics and geometry. By following a step-by-step approach and verifying the results, we can ensure the accuracy of our calculations and deepen our understanding of the subject matter. This exercise not only enhances our mathematical skills but also our problem-solving abilities, which are valuable in many aspects of life.