What is the rebound ratio as a percent? Write a model for their data and sketch a graph of the situation.
Understand the Problem
The question asks for the rebound ratio of a bouncing basketball based on provided data about its heights after each bounce. It also requires us to write a mathematical model for the data and to sketch a graph that represents the situation.
Answer
The rebound ratio is $80\%$, and the model is $y = 80 \times (0.8)^n$.
Answer for screen readers
The rebound ratio is approximately $80%$. The mathematical model for the bounce height is given by:
$$ y = 80 \times (0.8)^n $$
Steps to Solve
- Determine the rebound ratio
To find the rebound ratio, calculate how much the height decreases from one bounce to the next. Use the formula:
$$ \text{Rebound Ratio} = \frac{\text{Height after Bounce}}{\text{Height before Bounce}} $$
For example, the rebound ratio after the first bounce (from height 80 cm to 64 cm) is:
$$ \text{Rebound Ratio}_1 = \frac{64}{80} = 0.8 $$
- Calculate Rebound Ratios for Each Bounce
Now, find the rebound ratios for all bounces:
-
From 1st to 2nd: $$ \text{Rebound Ratio}_2 = \frac{51}{64} \approx 0.796875 \approx 0.80 $$
-
From 2nd to 3rd: $$ \text{Rebound Ratio}_3 = \frac{41}{51} \approx 0.803921557 \approx 0.80 $$
The rebound ratio is consistent and approximates to 0.8 or 80%.
- Write a Mathematical Model
The heights after each bounce can be represented as a model. The height after $n$ bounces (where the first height is $80$ cm) can be expressed as:
$$ y = 80 \times (0.8)^n $$
Where $y$ is the height in cm and $n$ is the bounce number.
- Sketch the Graph
To sketch the graph, plot the points:
- (1, 80)
- (2, 64)
- (3, 51)
- (4, 41)
Then draw a curve representing the function $y = 80 \times (0.8)^n$. The graph will show a decreasing exponential trend.
The rebound ratio is approximately $80%$. The mathematical model for the bounce height is given by:
$$ y = 80 \times (0.8)^n $$
More Information
The rebound ratio indicates how much of the original height the ball retains after each bounce. The model demonstrates that the height decreases exponentially, meaning that as the number of bounces increases, the height approaches zero but never quite reaches it.
Tips
- Confusing the rebound ratio with the height itself. Remember the rebound ratio represents the proportion of height retained after each bounce, not the height value.
- Not simplifying ratios correctly. Ensure to carry out the divisions and express them as decimals and percentages when needed.
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