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A bouncy ball is dropped such that the height of its first bounce is 3 feet and each successive bounce is 75% of the previous bounce's height. What would be the height of the 8th 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 feet and each successive bounce is 75% 75 \% of the previous bounce's height. What would be the height of the 88th 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 feet and each successive bounce is 75% 75 \% of the previous bounce's height. What would be the height of the 88th bounce of the ball? Round to the nearest tenth (if necessary).\newlineAnswer: □\square feet
  1. Understand the pattern: Understand the pattern of the bounces.\newlineThe height of each bounce is 75%75\% of the height of the previous bounce. This is a geometric sequence where each term is 0.750.75 times the previous term.
  2. Write formula for nth term: Write down 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 and the common ratio for this problem.\newlineThe first term a1a_1 is the height of the first bounce, which is 33 feet. The common ratio rr is 0.750.75, as each bounce is 75%75\% of the previous one.
  4. Calculate 88th bounce: Calculate the height of the 88th bounce using the formula.\newlineSubstitute a1=3a_1 = 3 feet and r=0.75r = 0.75 into the formula to find the 88th term (a8a_8).\newlinea8=3×0.758−1a_8 = 3 \times 0.75^{8-1}\newlinea8=3×0.757a_8 = 3 \times 0.75^7
  5. Perform calculation: Perform the calculation.\newlinea8=3×0.757a_8 = 3 \times 0.75^7\newlinea8=3×0.13348388671875a_8 = 3 \times 0.13348388671875\newlinea8≈0.40045166015625a_8 \approx 0.40045166015625 feet
  6. Round to nearest tenth: Round the result to the nearest tenth.\newlineThe height of the 88th bounce rounded to the nearest tenth is approximately 0.40.4 feet.

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