Final answer:
To find the height of the embankment formed by the earth taken out of a well that is 3 m in diameter and 14 m deep, and spread evenly to form a ring of 4 m width, we equate the volume of the well with the volume of the ring to solve for the embankment's height. After calculations, the height of the embankment turns out to be 2 m.
Step-by-step explanation:
To find the height of the embankment, we first assess the volume of the earth taken out from the well. The volume of the cylindrical well is given by the formula V = πr^2h, where r is the radius and h is the height. The well has a diameter of 3 m, so its radius is 1.5 m. The well is 14 m deep. So, the volume of earth taken from the well is V = π(1.5 m)^2(14 m).
The earth is used to form a circular ring around the well. The outer radius of the ring is the well radius plus the width of the ring, so it is 1.5 m + 4 m = 5.5 m. The inner radius of the ring is just the well radius, which is 1.5 m. The ring's height is what we want to find.
The volume of the circular ring (annulus) is equal to the volume of the earth taken out from the well and is given by V = π(h_{embankment})(R^2 - r^2), where R is the outer radius, r is the inner radius, and h_{embankment} is the height of the embankment.
Substituting the values, we get π(5.5^2 - 1.5^2)(h_{embankment}) = π(1.5)^2(14). We can cancel π on both sides and solve for h_{embankment}. By performing the calculations, we find that h_{embankment} is equal to 2 m. Hence, option (a) 2 m is the correct answer.