Calculating Embankment Height from Well Earth

This problem involves understanding the concept of volume conservation. When earth is dug out from a well, its volume is equal to the volume of the earth used to form the embankment. The well is cylindrical, and the embankment is in the shape of a cylindrical shell (a larger cylinder with a smaller cylinder removed from its center).

Understanding the Dimensions

• The well has a diameter of 3m, which means its radius is $r_{well} = 3m / 2 = 1.5m$.
• The well has a depth of 14m, so its height is $h_{well} = 14m$.
• The earth from the well forms an embankment in the shape of a circular ring around the well.
• The width of the embankment ring is 4m.
• The inner radius of the embankment ring is the same as the radius of the well, $r_{inner} = r_{well} = 1.5m$.
• The outer radius of the embankment ring is the inner radius plus the width: $r_{outer} = r_{inner} + width = 1.5m + 4m = 5.5m$.
• Let $h_{embankment}$ be the height of the embankment we need to find.

Volume of the Well (Earth Excavated)

The volume of a cylinder is given by the formula $V = \pi r^2 h$.

The volume of earth dug out from the well is:

$\qquad V_{well} = \pi \times (r_{well})^2 \times h_{well}$

$\qquad V_{well} = \pi \times (1.5m)^2 \times 14m$

$\qquad V_{well} = \pi \times 2.25 \times 14 \text{ m}^3$

$\qquad V_{well} = 31.5\pi \text{ m}^3$

Volume of the Embankment

The embankment is a cylindrical shell. Its volume is the volume of the larger cylinder (with radius $r_{outer}$) minus the volume of the smaller cylinder (with radius $r_{inner}$), both having the height $h_{embankment}$.

The volume of the embankment is:

$\qquad V_{embankment} = (\text{Volume of cylinder with radius } r_{outer}) - (\text{Volume of cylinder with radius } r_{inner})$

$\qquad V_{embankment} = (\pi \times (r_{outer})^2 \times h_{embankment}) - (\pi \times (r_{inner})^2 \times h_{embankment})$

$\qquad V_{embankment} = \pi \times h_{embankment} \times ((r_{outer})^2 - (r_{inner})^2)$

Substitute the values for $r_{outer}$ and $r_{inner}$:

$\qquad V_{embankment} = \pi \times h_{embankment} \times ((5.5m)^2 - (1.5m)^2)$

$\qquad V_{embankment} = \pi \times h_{embankment} \times (30.25 - 2.25) \text{ m}^2$

$\qquad V_{embankment} = \pi \times h_{embankment} \times 28 \text{ m}^2$

$\qquad V_{embankment} = 28\pi \times h_{embankment} \text{ m}^3$

Finding the Height of the Embankment

The volume of earth dug from the well is equal to the volume of the embankment formed.

$\qquad V_{well} = V_{embankment}$

$\qquad 31.5\pi = 28\pi \times h_{embankment}$

Now, we can cancel $\pi$ from both sides of the equation:

$\qquad 31.5 = 28 \times h_{embankment}$

To find $h_{embankment}$, divide 31.5 by 28:

$\qquad h_{embankment} = \frac{31.5}{28}$

To simplify the fraction, we can write 31.5 as 315/10:

$\qquad h_{embankment} = \frac{315/10}{28} = \frac{315}{10 \times 28} = \frac{315}{280}$

Divide the numerator and denominator by their greatest common divisor. Both are divisible by 35.

$\qquad 315 \div 35 = 9$

$\qquad 280 \div 35 = 8$

So, the height of the embankment is:

$\qquad h_{embankment} = \frac{9}{8} m$

Summary of Calculation Steps

1. Calculate the radius of the well.
2. Calculate the volume of the well using the cylinder volume formula.
3. Calculate the inner and outer radii of the embankment ring.
4. Set up the formula for the volume of the embankment (cylindrical shell).
5. Equate the volume of the well to the volume of the embankment.
6. Solve the resulting equation for the height of the embankment.
7. Simplify the final fraction.

Revision Table: Key Formulas & Values

Additional Information: Volumes of Solids

Understanding the formulas for volumes of basic 3D shapes is crucial for solving geometry problems like this. Here are some common volume formulas:

• Cylinder: The volume of a cylinder is the area of its base (a circle) multiplied by its height. $V = \pi r^2 h$, where $r$ is the radius of the base and $h$ is the height.
• Rectangular Prism: The volume is the product of its length, width, and height. $V = l \times w \times h$.
• Cone: The volume is one-third the volume of a cylinder with the same base and height. $V = \frac{1}{3} \pi r^2 h$.
• Sphere: The volume is given by $V = \frac{4}{3} \pi r^3$, where $r$ is the radius.
• Cylindrical Shell: The volume of a hollow cylinder or ring (like our embankment) can be found by subtracting the volume of the inner cylinder from the volume of the outer cylinder. $V = \pi R^2 h - \pi r^2 h = \pi h (R^2 - r^2)$, where $R$ is the outer radius, $r$ is the inner radius, and $h$ is the height.

In this well and embankment problem, we applied the concepts of cylinder volume and cylindrical shell volume, along with the principle of conservation of volume of earth.