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A bouncy ball is dropped such that the height of its first bounce is 3.75 feet and each successive bounce is 79% of the previous bounce's height. What would be the height of the 12th bounce of the ball? Round to the nearest tenth (if necessary). Answer: □ feet

A bouncy ball is dropped such that the height of its first bounce is 33.7575 feet and each successive bounce is 79% 79 \% of the previous bounce's height. What would be the height of the 1212th bounce of the ball? Round to the nearest tenth (if necessary).\newlineAnswer: □\square feet

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Full solution

Q. A bouncy ball is dropped such that the height of its first bounce is 33.7575 feet and each successive bounce is 79% 79 \% of the previous bounce's height. What would be the height of the 1212th bounce of the ball? Round to the nearest tenth (if necessary).\newlineAnswer: □\square feet
  1. Understand the pattern: Understand the pattern of the bounces. Each bounce is 79%79\% of the height of the previous bounce. This is a geometric sequence where each term is 79%79\% (or 0.790.79 times) the previous term.
  2. Write formula for nth term: Write the formula for the nth term of a geometric sequence.\newlineThe nth term ana_n of a geometric sequence can be found using the formula an=a1⋅r(n−1)a_n = a_1 \cdot r^{(n-1)}, where a1a_1 is the first term and rr is the common ratio.
  3. Identify first term and ratio: Identify the first term (a1a_1) and the common ratio (rr).\newlineThe first term a1a_1 is the height of the first bounce, which is 3.753.75 feet. The common ratio rr is 0.790.79, as each bounce is 79%79\% of the previous one.
  4. Calculate 1212th bounce height: Calculate the height of the 1212th bounce using the formula.\newlineSubstitute a1=3.75a_1 = 3.75 and r=0.79r = 0.79 into the formula to find the 1212th term (a12a_{12}).\newlinea12=3.75×0.7912−1a_{12} = 3.75 \times 0.79^{12-1}\newlinea12=3.75×0.7911a_{12} = 3.75 \times 0.79^{11}
  5. Perform the calculation: Perform the calculation.\newlinea12=3.75×0.7911a_{12} = 3.75 \times 0.79^{11}\newlinea12≈3.75×0.104975772a_{12} \approx 3.75 \times 0.104975772\newlinea12≈0.3936604a_{12} \approx 0.3936604
  6. Round the result: Round the result to the nearest tenth.\newlineThe height of the 1212th bounce is approximately 0.39366040.3936604 feet. Rounding to the nearest tenth gives us 0.40.4 feet.

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