Question

A ball Is released at a height of 8 inches to roll inside a half-cylinder. It rolls to a height of 2 inches on the other side of the cylinder on roll 1. Each time it rolls up a side of the cylinder, the ball reaches a point that is § as high as it had reached on the other side.
This explicit formula models the height of the ball, in inches, the rth time it rolls up a side of the cylinder.
How high does the ball roll on its 3rd time up the cylinder's side?
an = 2 • (1/4) ^(n—1)

Answers

GPT-4o mini
To find the height the ball reaches on its 3rd time up the cylinder's side, we will use the given explicit formula:

\[
a_n = 2 \cdot \left(\frac{1}{4}\right)^{n-1}
\]

Here, \(n\) represents the number of times the ball has rolled up the side of the cylinder. We need to find \(a_3\) (the height after the 3rd roll).

Plugging \(n = 3\) into the formula, we have:

\[
a_3 = 2 \cdot \left(\frac{1}{4}\right)^{3-1}
\]

Calculating the exponent:

\[
a_3 = 2 \cdot \left(\frac{1}{4}\right)^{2} = 2 \cdot \frac{1}{16}
\]

Now perform the multiplication:

\[
a_3 = \frac{2}{16} = \frac{1}{8}
\]

Thus, the height the ball reaches on its 3rd time up the cylinder's side is:

\[
\boxed{\frac{1}{8}} \text{ inches}
\]

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