91Ó°ÊÓ

As the ball is dropped from rest, its initial velocity (\(u\)) is 0, so we can remove the \(ut\) term from the equation. Using \(d_1\) for the distance covered in the first three seconds, we have: \[d_1 = \frac{1}{2}gt^2\] We are given that it takes 3 seconds for the ball to cover this distance, so we plug in the values for \(g\) and \(t\): \[d_1 = \frac{1}{2}(10\,\text{m/s}^2)(3\,\text{s})^2 = \frac{1}{2}(10)(9) = 45\,\text{m}\]

Now we need to find the total time taken by the ball to reach the ground, then we'll calculate the distance covered in the last second. Let's consider the velocity of the ball at the last second, it can be given by \(v = u + gt_n\) where \(t_n\) is the total time taken for the ball to reach the ground from the tower. \(0 = 0 + (10)t_n\) \(t_n = 0\) As we know, \(t_n\) can't be zero, because the ball is moving. So, let's rewrite the velocity equation as \(v^2 = u^2 + 2gd\) Now, the distance (\(d\)) covered in the last second will be equal to the total distance (\(h\)) minus the distance covered before the last second (\(h - d_1\)). \[(v^2) = 2g(h - d_1)\]

In the problem statement, we are given that distance covered in the last second is equal to far it covered in the first 3 seconds, \(d_1\). Based on this condition, \((h - d_1) = d_1\) Thus, \(v^2 = 2g(2d_1)\) We substitute the value of \(d_1\) from step 1: \(v^2 = 2(10)(2(45))\) \(v^2 = 1800\) Now, we can use the above velocity to find the total height \(h\) Let's consider the equation \(v^2 = u^2 + 2gh\) Since the ball is dropped from rest, \(u = 0\). So, $h = \frac{v^2}{2g} = \frac{1800}{2(10)} = 90\,\text{m}\] However, none of the options matches 90 m. It's due to the incorrect interpretation of the problem. In the problem, it's mentioned that the distance covered in the last second is equal to the distance covered in the first three seconds. So, let's consider the correct interpretation.

The total distance traveled in time \(t_n - 1\) to \(t_n\) is equal to distance covered in the first three seconds. Let \(d_2\) be the distance traveled in the last second. We know, \(d_1 = 45\,\text{m}\), and \(d_2 = d_1 = 45\,\text{m}\) Now, let's utilize the equations of motion: \(h = d_1 + d_2\) $h = 45 + 45 = 90\,\text{m}\] Thus, the correct height of the tower is not available in the given options. However, by understanding the problem thoroughly and using the correct interpretation, we find that the height of the tower is \(90\,\text{m}\).