Chapter 1: Answers 4 Jack K. Cohen Colorado School of Mines

  1. r $\approx$ 50, 008.33. A real pitfall here is getting trapped into investigating the regime where r < s = 1000—``obviously'' the radius of the asteroid would sensibly be much larger than s. By the way, the word ``obviously'' in mathematics means ``clear when you think about it for awhile''. For example, some real smart people have been known to have put up a first Plot for values of r near 0 in solving the present problem.

    Another problem is that we have no idea where to hunt for the ``physically sensible'' root. As promised in the problem statement, we will develop methods for this in a few weeks.

  2. (1.2.24) A sketch reveals some nice right triangles: d = $\sqrt{{2}}$.

  3. It checks.

  4. No solution given to honor problems.

  5. (1.2.25) d = ${\frac{{4}}{{13}}}$ $\sqrt{{26}}$.

    1. It checks (despite the answer in the back of the book).
    2. The form y = mx + b1, y = mx + b2 doesn't include the case of two vertical lines.

    1. y = 3x - 2.

    2. y = 2.1x - 1.1.

    3. y = 2.01x - 1.01.

    4. Slope is m = 2, tangent line is y = 2x - 1.

    5. Did you notice that the curve for y = x2 was almost linear in the ``zoomed'' plot?

    6. The precise definition requires calculus concepts that will be discussed soon. (Attempts to give a definition in terms of only algebra and geometry concepts are sure to be inadequate.) It is a secant line.