Questions: A desalination process is removing salt from seawater. A scientist collects the following data after observing the amount of salt in a seawater sample. Time (minutes) 25 50 75 100 125 150 175 200 Amount of salt (grams) 257 140 78 55 28 14 11 3 The following exponential model predicts the amount A of salt t minutes after the process has begun. A=451.008(0.978)^t Why was an exponential function chosen to model this data? An exponential function was chosen because the amount of salt is decreasing less and less each minute. Use the model to determine the amount of salt in the seawater (in grams) 50 minutes after the process had begun. (Round your answer to the nearest whole number.) × grams How does the predicted answer from the model compare to the data collected from the observations? The model the amount of salt by x grams.

A desalination process is removing salt from seawater. A scientist collects the following data after observing the amount of salt in a seawater sample. Time (minutes) 25 50 75 100 125 150 175 200 Amount of salt (grams) 257 140 78 55 28 14 11 3 The following exponential model predicts the amount A of salt t minutes after the process has begun. A=451.008(0.978)^t Why was an exponential function chosen to model this data? An exponential function was chosen because the amount of salt is decreasing less and less each minute. Use the model to determine the amount of salt in the seawater (in grams) 50 minutes after the process had begun. (Round your answer to the nearest whole number.) × grams How does the predicted answer from the model compare to the data collected from the observations? The model the amount of salt by x grams.
Transcript text: A desalination process is removing salt from seawater. A scientist collects the following data after observing the amount of salt in a seawater sample. \begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline Time (minutes) & 25 & 50 & 75 & 100 & 125 & 150 & 175 & 200 \\ \hline Amount of salt (grams) & 257 & 140 & 78 & 55 & 28 & 14 & 11 & 3 \\ \hline \end{tabular} The following exponential model predicts the amount $A$ of salt $t$ minutes after the process has begun. \[ A=451.008(0.978)^{t} \] Why was an exponential function chosen to model this data? An exponential function was chosen because the amount of salt is decreasing less and less each minute. Use the model to determine the amount of salt in the seawater (in grams) 50 minutes after the process had begun. (Round your answer to the nearest whole number.) $\square$ $\times$ grams How does the predicted answer from the model compare to the data collected from the observations? The model $\square$ the amount of salt by $\square$ $x$ grams.
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Solution

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Solution Steps

Step 1: Exponential Model Prediction

The exponential model predicting the amount of salt \( A \) in grams after \( t \) minutes is given by:

\[ A = 451.008 \cdot (0.978)^t \]

To find the amount of salt at \( t = 50 \) minutes, we substitute \( t \) into the equation:

\[ A = 451.008 \cdot (0.978)^{50} \]

Calculating this gives:

\[ A \approx 148 \text{ grams} \]

Step 2: Comparison with Observed Data

The observed amount of salt at \( t = 50 \) minutes is \( 140 \) grams. We compare the predicted amount with the observed amount:

\[ \text{Difference} = 148 - 140 = 8 \text{ grams} \]

Step 3: Interpretation of Results

Since the predicted amount of salt is greater than the observed amount, we conclude that the model overestimates the amount of salt.

Final Answer

The predicted amount of salt at \( t = 50 \) minutes is \( \boxed{148} \) grams, and the model overestimates the amount of salt by \( \boxed{8} \) grams.

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