somthing like this ?
for the b17 and lancs trying to get 30k alt
3.8 Project
How Deep Does a Floating Ball Sink?
Figure 3.8.24 in the text shows a large cork ball of radius a = 1 floating in water of density 1. Let the ball's density be denoted by r. If r = 1/4, then Archimedes' law of buoyancy implies that the ball floats in such a way that one-fourth of its total volume is submerged. Because the volume of the whole ball of radius 1 is 4p/3, it follows that the volume of the part of the ball beneath the waterline is given by
(1)
The shape of the submerged part of the ball is that of a spherical segment with a circular flat top. The volume of a spherical segment of top radius r and depth h = x (as in the figure) is given by the known formula
(2)
This formula is also due to Archimedes and holds for any depth x, whether the spherical segment is less than or greater than a hemisphere. For instance, note that with r = 0 and
x = 2a it gives V = 4 p a3/3, the volume of a whole sphere of radius a.
For a preliminary investigation, proceed as follows to find the depth x to which the ball sinks in the water. Equate the two expressions for V in (1) and (2), and then use the Pythagorean formula for the right triangle in the figure to eliminate r. You should find that x must be a solution of the cubic equation
(3)
As the graph for –4 £ x £ 4 in Fig. 3.8.25 in the text indicates, this equation has three real solutions — one in the interval (–1, 0), one in (0, 1), and one in (2, 3). The solution between 0 and 1 gives the actual depth x to which the ball sinks (why?). You can proceed to find this value of x using Newton's method.
Your Investigation: For your very own floating ball to investigate, let its density r in Eq. (1) be given by
where k denotes the last nonzero digit in the sum of the final 4 digits of your student ID number. Your objective is to find the depth to which this ball sinks in the water. Start by deriving the cubic equation that you need to solve, explaining each step carefully. Then find each of its solutions accurate to at least four decimal places.
