Question

1. a ball is dropped from the top of a building. the table shows its height in feet above ground at the top of each bounce. what is the height of the ball at the top of bounce 6?

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$$\begin{tabular}{|c|c|} \\hline bounce & height \\\\ \\hline 1 & 250 \\\\ \\hline 2 & 200 \\\\ \\hline 3 & 160 \\\\ \\hline 4 & 128 \\\\ \\hline 5 & 102.4 \\\\ \\hline \\end{tabular}$$

Explanation:

Identify the sequence type

The table shows the height of a ball after each bounce:

• Bounce 1: \(250\) feet
• Bounce 2: \(200\) feet
• Bounce 3: \(160\) feet
• Bounce 4: \(128\) feet
• Bounce 5: \(102.4\) feet

We check if this is a Geometric Sequence by finding the ratio between consecutive terms.

Find the common ratio

Using the Common Ratio concept:

$$ r = \frac{200}{250} = 0.8 $$

$$ r = \frac{160}{200} = 0.8 $$

$$ r = \frac{128}{160} = 0.8 $$

$$ r = \frac{102.4}{128} = 0.8 $$

The common ratio is \(r = 0.8\).

Calculate the next term

Using the Geometric Sequence Terms concept, the height at bounce 6 is:

$$ a_6 = a_5 \times r $$

$$ a_6 = 102.4 \times 0.8 $$

$$ a_6 = 81.92 $$

Answer:

\(81.92\) feet