A bouncing ball rises to 60% of the height from which it fell. If the ball fell from a 10 foot balcony, express the height "h" in feet as a function of the number of bounces "n".
height after 1st bounce = 10(.6)
height after 2nd bounce = (10(.6))(.6) = 10(.6^2)
height after 3rd bounce = 10(.6^3)
...
height after nth bounce = 10(.6^n)
To express the height "h" as a function of the number of bounces "n", we need to determine how the height changes after each bounce.
Given that the ball rises to 60% of the height from which it fell, we can calculate the height of the ball after each bounce. Let's break down the problem step by step:
1. Start by setting the initial height of the ball, h₀, which is when it falls from the 10-foot balcony. Therefore, h₀ = 10.
2. After the first bounce, the ball rises to 60% of the height it was at before bouncing. This can be expressed as h₁ = 0.6 * h₀.
3. After the second bounce, the ball reaches 60% of the height it was at after the first bounce. This can be expressed as h₂ = 0.6 * h₁.
4. Similarly, after the third bounce, the height can be expressed as h₃ = 0.6 * h₂, and so on.
In general, we can express the height "h" after the nth bounce as:
hₙ = 0.6 * hₙ₋₁
Now let's write the expression for h in terms of n:
hₙ = 0.6 * hₙ₋₁
Since hₙ₋₁ = 0.6 * hₙ₋₂, we can substitute it into the equation:
hₙ = 0.6 * (0.6 * hₙ₋₂)
By continuing this substitution process for each previous bounce, we can find an expression for h in terms of n:
hₙ = 0.6ⁿ * h₀
Using the initial height h₀ = 10, the expression for height "h" as a function of the number of bounces "n" is:
h(n) = 0.6ⁿ * 10
Therefore, the height "h" in terms of the number of bounces "n" is given by the function h(n) = 0.6ⁿ * 10.