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IB Mathematics: Applications and Interpretation HL · 12.1: Sequences & Series · 12.1.9

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Infinite series in context: bouncing balls and nested squares Problem set B

Work out the first few terms from the words of the question. Check what the first term is and where counting starts.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. A super-ball is dropped from the top of a climbing wall 9 m high. After each bounce it rises to 70% of the height of the bounce before. Find the height it reaches after the 7th bounce.

  2. A steel bead is dropped onto a metal plate and takes 3 s to fall. Each time it bounces it takes 60% of the time of the previous fall to rise, and the same time to fall back.

    1. (a) Show that the total time of the motion is given by \(3+6(0.6)+6(0.6)^2+\dots\) seconds.
    2. (b) Find a formula for \(S_n\), the total time of the first n terms.

  3. A steel bead is dropped onto a metal plate and takes 3 s to fall. Each time it bounces it takes 60% of the time of the previous fall to rise, and the same time to fall back. Find the time from the drop until the bead comes to rest.

  4. Given that \(x+\frac{x^2}{4}+\frac{x^3}{16}+\dots=2\), find the value of x.

In context

  1. A basketball is dropped from a height of 8 m. After each bounce it rises to 85% of the height of the bounce before. Find the total distance it travels before it comes to rest.

  2. A square tile has sides of 14 cm. The midpoints of its sides are joined to make a new square, and the two triangles at the bottom-left and top-right corners are shaded. The same is done in the new square, and again, as shown. Find the total shaded area after 6 stages.

  3. A square tile has sides of 14 cm. The midpoints of its sides are joined to make a new square, and the two triangles at the bottom-left and top-right corners are shaded. The same is done for ever. Find the total shaded area.

Stretch

  1. A ball is dropped and bounces. The total time of the first n terms of its motion is \(S_n=12-9(0.6)^{n-1}\) seconds. Find the least value of n for which \(S_n>11.5\).

jmaths.xyz · AI HL · 12.1.9 Infinite series in context: bouncing balls and nested squares