Question

A well of 3 meter in diameter is dug to a depth of 14 meter. An embankment is made by spreading the soil out of it, making a circular ring of 4 meters wide, evenly around the well. Find the height of the embankment.

Hint:

To find the height of an embankment formed by spreading soil from a well, calculate the volume of the soil removed and divide it by the area of the embankment ring.

Answer 1

Step 1: Calculate the volume of the soil removed from the well.
The well is in the shape of a cylinder with a radius \( r_1 = \frac{3}{2} = 1.5 \) meters and a depth (height) of \( h_1 = 14 \) meters. The volume of the soil removed from the well is the volume of this cylinder, given by the formula: \[ V_{\text{well}} = \pi r_1^2 h_1 \] Substituting the values: \[ V_{\text{well}} = \pi (1.5)^2 \times 14 = \pi \times 2.25 \times 14 = 31.5 \pi \, \text{cubic meters.} \]
Step 2: Calculate the area of the embankment.
The embankment forms a circular ring around the well. The inner radius of the embankment is \( r_1 = 1.5 \) meters and the outer radius is \( r_2 = 1.5 + 4 = 5.5 \) meters. The area of the ring (embankment) is the area of the outer circle minus the area of the inner circle: \[ A_{\text{embankment}} = \pi r_2^2 - \pi r_1^2 = \pi \left( (5.5)^2 - (1.5)^2 \right) \] Calculating the areas: \[ A_{\text{embankment}} = \pi \left( 30.25 - 2.25 \right) = \pi \times 28 = 28 \pi \, \text{square meters.} \]
Step 3: Find the height of the embankment.
The height \( h_2 \) of the embankment can be found using the formula for the volume of a cylinder, \( V = A \times h \), where \( A \) is the area and \( h \) is the height. The volume of the embankment is equal to the volume of the soil removed from the well, so: \[ V_{\text{embankment}} = A_{\text{embankment}} \times h_2 = 31.5 \pi \] Substituting the area of the embankment: \[ 28 \pi \times h_2 = 31.5 \pi \] Dividing both sides by \( 28 \pi \): \[ h_2 = \frac{31.5}{28} = 1.125 \, \text{meters.} \]
Conclusion:
The height of the embankment is \( \boxed{1.125} \, \text{meters.} \)