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CS13002 PDS Lab, Spring 2003, Section 1/A

Assignment 1

(Unless otherwise stated use integer arithmetic)

**[Warm-up exercise]** (Not to be submitted)

Show that -29 and 31 are roots of the polynomial

X^{3} + X^{2} - 905X - 2697.

What is its third root?

**[Warm-up exercise]** (Not to be submitted)

Show that -2931 is a root of the polynomial

X^{3} + 2871X^{2} - 174961X + 2634969.

**[First execise]**

The three roots of the polynomial

X^{3} + X^{2} - 74034X + 5294016

are integers. Find them.

**[Bonus execise]**

The three roots of the polynomial

X^{3} + X^{2} - 28033X - 1815937

are again integers. Find them.

**[Second execise]**

**[The queer rope and the diligent ant]**
An ant is sitting at the left end of a rope of length
10 cm. At t=0 the ant starts moving along the rope
to reach the other end of the rope. The ant has a speed
of 1 cm per second. After every second the rope stretches
instantaneously and uniformly (along its length) by 10 cm
with the left end fixed at the point from where the ant
started its journey. Suppose that the ant's legs provide it
sufficient friction in order to withstand the stretching
of the rope without slipping.
Write a program to demonstrate that
the ant will be able to reach the right end of the rope.
Your program should also calculate how many seconds the ant
would take to achieve this goal. You may assume that the
length of the ant is negligible (i.e., zero).

**Note:** Use real (`float` or `double`) arithmetic.

**Note:** The ant would reach the
right end of the rope, even if its initial length and stretching
per second were 1 km (or even a billion kilometers) instead of
10 cm. But for these dimensions the ant would take such an unbelievably
large time that your program will not give you the confirmation in your
life-time. Moreover, you will require more precision than what `double`
can provide. Try solving this puzzle mathematically.

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