A ball is dropped from an initial height of 22 feet and bounces off the ground repeatedly. The function h...

1. TRANSLATE the problem information

• Given information:
  • Initial height: 22 feet
  • Each bounce reaches 85% of previous maximum height
  • Need \(\mathrm{h(n)}\) = height after n ground hits

• What this tells us:
  • 85% means multiply by 0.85 each time
  • We start with 22 feet

2. INFER the mathematical pattern

• This is an exponential decay situation because:
  • We repeatedly multiply by the same factor (0.85)
  • The height gets smaller each time \(\mathrm{(0.85\ \lt\ 1)}\)

• Pattern recognition:
  • After 1 hit: \(\mathrm{22\ \times\ 0.85}\)
  • After 2 hits: \(\mathrm{22\ \times\ 0.85\ \times\ 0.85\ =\ 22\ \times\ (0.85)^2}\)
  • After n hits: \(\mathrm{22\ \times\ (0.85)^n}\)

3. TRANSLATE into function form

• The exponential function format is \(\mathrm{h(n)\ =\ a(r)^n}\) where:
  • \(\mathrm{a}\) = initial value = 22
  • \(\mathrm{r}\) = common ratio = 0.85
  • \(\mathrm{n}\) = number of bounces

• Therefore: \(\mathrm{h(n)\ =\ 22(0.85)^n}\)

Answer: B. \(\mathrm{h(n)\ =\ 22(0.85)^n}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse what the initial value should be, thinking the coefficient should be 85 instead of 22.

They reason: "Since each bounce is 85% of the previous, the function should start with 85." This misses that 85% is the multiplier (0.85), while 22 feet is the actual starting height from which everything scales.

This may lead them to select Choice D (\(\mathrm{h(n)\ =\ 85(0.85)^n}\))

Second Most Common Error:

Poor TRANSLATE reasoning: Students convert 85% incorrectly, thinking it should be 0.22 instead of 0.85.

They might subtract from 1 incorrectly: "The ball loses 15% each time, so that's 85% - 100% = -15%... wait, that's not right... maybe it's 100% - 85% = 15% = 0.15... no, that's still wrong... maybe 0.22?" This confusion about percentage conversion creates calculation errors.

This may lead them to select Choice A (\(\mathrm{h(n)\ =\ 22(0.22)^n}\))

The Bottom Line:

This problem requires precise translation of percentage language into mathematical operations. Students must clearly distinguish between the decay factor \(\mathrm{(85\%\ =\ 0.85)}\) and the initial height (22 feet), then recognize the exponential pattern that emerges.