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Answer :
To find the radical function that best models the increase in the number of women physicians, we will compare each function option with the actual data provided. The model that produces function values closest to the data is the best fit.
Let's break down each step:
1. Understand the Data: We have data showing the increase in the number of women physicians (in thousands) from different years. The years are given as years after 1960:
- 1 year (1961): 1.7 thousand
- 4 years (1964): 13.6 thousand
- 10 years (1970): 53.7 thousand
- 13 years (1973): 79.6 thousand
- 17 years (1977): 119.1 thousand
2. Function Options: Each option given is a function that involves calculating a root or power:
- A. [tex]\( f(x) = 1.69 \cdot \sqrt[5]{x^3} \)[/tex]
- B. [tex]\( f(x) = 1.69 \cdot \sqrt[2]{x^5} \)[/tex]
- C. [tex]\( f(x) = 1.69 \cdot \sqrt[2]{x^3} \)[/tex]
- D. [tex]\( f(x) = 1.69 \cdot \sqrt[3]{x^2} \)[/tex]
3. Evaluation of Models: Let's calculate the predicted values for each function and compare them with the actual data:
- Model A: Predicts values that are quite different from the actual values, leading to a high discrepancy.
- Model B: Results in predictions that are excessively high compared to the actual data.
- Model C: Produces values that are very close to what's observed:
- For 1 year: approximately 1.69 thousand
- For 4 years: approximately 13.52 thousand
- For 10 years: approximately 53.44 thousand
- For 13 years: approximately 79.21 thousand
- For 17 years: approximately 118.46 thousand
- Model D: Predicts values much lower than the actual data.
4. Best Fit: Model C provides predictions that closely match the actual values given in the table. The predicted values are nearly identical to the observed data points, indicating that Model C is a suitable choice.
Therefore, the radical function that best models the increase of women physicians is:
C. [tex]\( f(x) = 1.69 \cdot \sqrt[2]{x^3} \)[/tex]
Let's break down each step:
1. Understand the Data: We have data showing the increase in the number of women physicians (in thousands) from different years. The years are given as years after 1960:
- 1 year (1961): 1.7 thousand
- 4 years (1964): 13.6 thousand
- 10 years (1970): 53.7 thousand
- 13 years (1973): 79.6 thousand
- 17 years (1977): 119.1 thousand
2. Function Options: Each option given is a function that involves calculating a root or power:
- A. [tex]\( f(x) = 1.69 \cdot \sqrt[5]{x^3} \)[/tex]
- B. [tex]\( f(x) = 1.69 \cdot \sqrt[2]{x^5} \)[/tex]
- C. [tex]\( f(x) = 1.69 \cdot \sqrt[2]{x^3} \)[/tex]
- D. [tex]\( f(x) = 1.69 \cdot \sqrt[3]{x^2} \)[/tex]
3. Evaluation of Models: Let's calculate the predicted values for each function and compare them with the actual data:
- Model A: Predicts values that are quite different from the actual values, leading to a high discrepancy.
- Model B: Results in predictions that are excessively high compared to the actual data.
- Model C: Produces values that are very close to what's observed:
- For 1 year: approximately 1.69 thousand
- For 4 years: approximately 13.52 thousand
- For 10 years: approximately 53.44 thousand
- For 13 years: approximately 79.21 thousand
- For 17 years: approximately 118.46 thousand
- Model D: Predicts values much lower than the actual data.
4. Best Fit: Model C provides predictions that closely match the actual values given in the table. The predicted values are nearly identical to the observed data points, indicating that Model C is a suitable choice.
Therefore, the radical function that best models the increase of women physicians is:
C. [tex]\( f(x) = 1.69 \cdot \sqrt[2]{x^3} \)[/tex]
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