Calculating the Volume of the Well

Oct 13, 2025 by miodeya.com 133 views

Introduction

In this problem, we are given a well with a diameter of 7 m that has been dug 45 m deep. The earth dug out is used to form an embankment around the well. We are asked to find the width of the embankment, given that its height is 3.0 m. This problem involves the concept of volume and the use of mathematical formulas to solve it.

Understanding the Problem

To solve this problem, we need to understand the concept of volume and how it relates to the dimensions of the well and the embankment. The volume of the earth dug out from the well is equal to the volume of the embankment. We can use this concept to set up an equation and solve for the width of the embankment.

Calculating the Volume of the Well

The volume of the well can be calculated using the formula for the volume of a cylinder:

V = πr^2h

where V is the volume, π is a constant approximately equal to 3.14, r is the radius of the well, and h is the height of the well.

Given that the diameter of the well is 7 m, the radius is half of the diameter, which is 3.5 m. The height of the well is 45 m. Plugging these values into the formula, we get:

V = π(3.5)^2(45) V = 3.14(12.25)(45) V = 1725.37 m^3

Calculating the Volume of the Embankment

The volume of the embankment is equal to the volume of the earth dug out from the well. We can use the same formula for the volume of a cylinder to calculate the volume of the embankment:

V = πr^2h

where V is the volume, π is a constant approximately equal to 3.14, r is the radius of the embankment, and h is the height of the embankment.

Given that the height of the embankment is 3.0 m, we need to find the radius of the embankment. Let's call the radius of the embankment "R". We can set up an equation using the formula for the volume of a cylinder:

V = πR^2(3.0)

We know that the volume of the embankment is equal to the volume of the well, which is 1725.37 m^3. We can set up an equation:

πR^2(3.0) = 1725.37

Solving for the Radius of the Embankment

To solve for the radius of the embankment, we can divide both sides of the equation by 3.0:

πR^2 = 1725.37 / 3.0 πR^2 = 575.12

Next, we can divide both sides of the equation by π:

R^2 = 575.12 / π R^2 = 183.45

Finally, we can take the square root of both sides of the equation to solve for the radius of the embankment:

R = √183.45 R = 13.55 m

Calculating the Width of the Embankment

The width of the embankment is equal to the difference between the radius of the embankment and the radius of the well. We can calculate the width of the embankment as follows:

Width = R - 3.5 Width = 13.55 - 3.5 Width = 10.05 m

Therefore, the width of the embankment is approximately 10.05 m.

Conclusion

In this problem, we used the concept of volume and mathematical formulas to solve for the width of the embankment. We calculated the volume of the well and the embankment, and then used the formula for the volume of a cylinder to solve for the radius of the embankment. Finally, we calculated the width of the embankment by subtracting the radius of the well from the radius of the embankment. The width of the embankment is approximately 10.05 m.

Introduction

In our previous article, we solved a problem involving a well with a diameter of 7 m that has been dug 45 m deep. The earth dug out is used to form an embankment around the well. We calculated the width of the embankment to be approximately 10.05 m. In this article, we will answer some frequently asked questions related to this problem.

Q: What is the formula for the volume of a cylinder?

A: The formula for the volume of a cylinder is V = πr^2h, where V is the volume, π is a constant approximately equal to 3.14, r is the radius of the cylinder, and h is the height of the cylinder.

Q: How do we calculate the volume of the well?

A: To calculate the volume of the well, we use the formula V = πr^2h, where r is the radius of the well and h is the height of the well. Given that the diameter of the well is 7 m, the radius is half of the diameter, which is 3.5 m. The height of the well is 45 m. Plugging these values into the formula, we get V = π(3.5)^2(45).

Q: What is the relationship between the volume of the well and the volume of the embankment?

A: The volume of the embankment is equal to the volume of the earth dug out from the well. This means that the volume of the embankment is equal to the volume of the well.

Q: How do we calculate the radius of the embankment?

A: To calculate the radius of the embankment, we use the formula V = πR^2(3.0), where R is the radius of the embankment and V is the volume of the embankment. We know that the volume of the embankment is equal to the volume of the well, which is 1725.37 m^3. We can set up an equation: πR^2(3.0) = 1725.37.

Q: What is the formula for the width of the embankment?

A: The width of the embankment is equal to the difference between the radius of the embankment and the radius of the well. We can calculate the width of the embankment as follows: Width = R - 3.5.

Q: What is the width of the embankment?

A: The width of the embankment is approximately 10.05 m.

Q: What are the key concepts involved in solving this problem?

A: The key concepts involved in solving this problem are the formula for the volume of a cylinder, the relationship between the volume of the well and the volume of the embankment, and the formula for the width of the embankment.

Q: What are the steps involved in solving this problem?

A: The steps involved in solving this problem are:

Calculate the volume of the well using the formula V = πr^2h.
Set up an equation using the formula V = πR^2(3.0) to calculate the radius of the embankment.
Solve for the radius of the embankment.
Calculate the width of the embankment using the formula Width = R - 3.5.

Conclusion

In this article, we answered some frequently asked questions related to the problem of a well with a diameter of 7 m that has been dug 45 m deep. The earth dug out is used to form an embankment around the well. We calculated the width of the embankment to be approximately 10.05 m. We hope that this article has provided a clear understanding of the key concepts and steps involved in solving this problem.