Math  /  Data & Statistics

Question```latex \text{Predict the ball's rebound height after each successive bounce if its starting height is 200 cm. Create a table with these predicted heights.}
\begin{tabular}{|c|c|} \hline \text{Bounce} & \text{Height (cm)} \\ \hline 1 & 166.4 \\ \hline 2 & 138.445 \\ \hline 3 & \\ \hline 4 & \\ \hline 5 & \\ \hline 6 & \\ \hline \end{tabular}
\text{If the ball was left to bounce uninterrupted how high ................} ```
\text{The user took a picture with their phone and the text was extracted above. The user then had a dialogue with an AI Assistant to help clarify the instructions.}
\text{Dialogue Transcript:}
\text{Hello! It looks like you have a question about predicting the rebound height of a ball. The information you've provided shows the ball starts at a height of 200 cm and gives the rebound heights for the first two bounces. However, there is missing information, and it seems the table is incomplete.}
\text{To help you further, I need to know:} \begin{enumerate} \item \text{The rebound ratio or the percentage of the height retained after each bounce.} \item \text{How many bounces you would like to calculate the height for.} \end{enumerate}
\text{Once you provide this information, I can assist you in completing the table with the predicted heights for each bounce. Looking forward to your response!}
\text{Extracted text from attached image:} \begin{tabular}{|c|c|} \hline \text{Drop Height (cm)} & \text{Rebound Height (cm)} \\ \hline 150 & 124 \\ 70 & 59 \\ 120 & 100 \\ 100 & 83 \\ 110 & 92 \\ 40 & 33 \\ \hline \end{tabular}

Studdy Solution

STEP 1

1. The initial height of the ball is 200 cm.
2. The rebound heights for the first two bounces are given: 166.4 cm and 138.445 cm.
3. The rebound ratio is consistent for each bounce.
4. The rebound ratio can be calculated from the given heights.
5. We need to predict the heights for bounces 3 to 6.

STEP 2

1. Calculate the rebound ratio from the given heights.
2. Use the rebound ratio to predict the heights for the remaining bounces.
3. Complete the table with the predicted heights.

STEP 3

Calculate the rebound ratio using the first bounce:
Rebound Ratio=Height after 1st bounceInitial Height=166.4200 \text{Rebound Ratio} = \frac{\text{Height after 1st bounce}}{\text{Initial Height}} = \frac{166.4}{200}
Rebound Ratio=0.832 \text{Rebound Ratio} = 0.832

STEP 4

Verify the rebound ratio using the second bounce:
Rebound Ratio=Height after 2nd bounceHeight after 1st bounce=138.445166.4 \text{Rebound Ratio} = \frac{\text{Height after 2nd bounce}}{\text{Height after 1st bounce}} = \frac{138.445}{166.4}
Rebound Ratio0.832 \text{Rebound Ratio} \approx 0.832
The calculated rebound ratio is consistent.

STEP 5

Predict the height for the 3rd bounce:
Height after 3rd bounce=Height after 2nd bounce×Rebound Ratio \text{Height after 3rd bounce} = \text{Height after 2nd bounce} \times \text{Rebound Ratio}
Height after 3rd bounce=138.445×0.832 \text{Height after 3rd bounce} = 138.445 \times 0.832
Height after 3rd bounce115.2 \text{Height after 3rd bounce} \approx 115.2

STEP 6

Predict the height for the 4th bounce:
Height after 4th bounce=Height after 3rd bounce×Rebound Ratio \text{Height after 4th bounce} = \text{Height after 3rd bounce} \times \text{Rebound Ratio}
Height after 4th bounce=115.2×0.832 \text{Height after 4th bounce} = 115.2 \times 0.832
Height after 4th bounce95.8 \text{Height after 4th bounce} \approx 95.8

STEP 7

Predict the height for the 5th bounce:
Height after 5th bounce=Height after 4th bounce×Rebound Ratio \text{Height after 5th bounce} = \text{Height after 4th bounce} \times \text{Rebound Ratio}
Height after 5th bounce=95.8×0.832 \text{Height after 5th bounce} = 95.8 \times 0.832
Height after 5th bounce79.7 \text{Height after 5th bounce} \approx 79.7

STEP 8

Predict the height for the 6th bounce:
Height after 6th bounce=Height after 5th bounce×Rebound Ratio \text{Height after 6th bounce} = \text{Height after 5th bounce} \times \text{Rebound Ratio}
Height after 6th bounce=79.7×0.832 \text{Height after 6th bounce} = 79.7 \times 0.832
Height after 6th bounce66.3 \text{Height after 6th bounce} \approx 66.3

STEP 9

Complete the table with the predicted heights:
\begin{tabular}{|c|c|} \hline \text{Bounce} & \text{Height (cm)} \\ \hline 1 & 166.4 \\ \hline 2 & 138.445 \\ \hline 3 & 115.2 \\ \hline 4 & 95.8 \\ \hline 5 & 79.7 \\ \hline 6 & 66.3 \\ \hline \end{tabular}

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