Answer :
Let's break down and solve each part of the problem step-by-step using the model provided, [tex]\( y = 2.9 \sqrt{x} + 36 \)[/tex].
### a. Head Circumference at Birth
Given [tex]\( x = 0 \)[/tex]:
1. Substitute [tex]\( x \)[/tex] into the equation:
[tex]\[ y = 2.9 \sqrt{0} + 36 \][/tex]
2. Simplify the equation:
[tex]\[ y = 2.9 \times 0 + 36 = 36 \][/tex]
3. Therefore, the head circumference at birth ([tex]\( x = 0 \)[/tex]) is:
[tex]\[ \boxed{36.0 \text{ cm}} \][/tex]
### b. Head Circumference at 9 Months
Given [tex]\( x = 9 \)[/tex]:
1. Substitute [tex]\( x \)[/tex] into the equation:
[tex]\[ y = 2.9 \sqrt{9} + 36 \][/tex]
2. Simplify the equation:
[tex]\[ y = 2.9 \times 3 + 36 = 8.7 + 36 = 44.7 \][/tex]
3. Therefore, the head circumference at 9 months ([tex]\( x = 9 \)[/tex]) is:
[tex]\[ \boxed{44.7 \text{ cm}} \][/tex]
### c. Head Circumference at 14 Months
Given [tex]\( x = 14 \)[/tex]:
1. Substitute [tex]\( x \)[/tex] into the equation:
[tex]\[ y = 2.9 \sqrt{14} + 36 \][/tex]
2. Simplify the equation:
[tex]\[ y \approx 2.9 \times 3.74 + 36 \approx 10.846 + 36 = 46.9 \][/tex]
3. Therefore, the head circumference at 14 months ([tex]\( x = 14 \)[/tex]) is:
[tex]\[ \boxed{46.9 \text{ cm}} \][/tex]
### d. Validating the Model
To compare the model with empirical data on your graphing calculator:
1. Plot the equation [tex]\( y = 2.9 \sqrt{x} + 36 \)[/tex] on a graphing calculator.
2. Locate the points for [tex]\( x = 0, 9, \)[/tex] and [tex]\( 14 \)[/tex] months on the graph.
3. Verify if the calculated values from this model match the points on the graph.
From the calculations:
- At [tex]\( x = 0 \)[/tex], the graph should show [tex]\( (0, 36) \)[/tex].
- At [tex]\( x = 9 \)[/tex], the graph should show [tex]\( (9, 44.7) \)[/tex].
- At [tex]\( x = 14 \)[/tex], the graph should show [tex]\( (14, 46.9) \)[/tex].
By comparing the points on the graph with the model-derived values, you can determine if the model provides an accurate representation of the head circumference over time. If the points match closely, then this is a good model. Given that models in science and mathematics often have small deviations from actual measurements, slight differences are expected. However, if the model-derived values are close to the plotted points, then this model is likely reliable.
#### Conclusion
The values obtained from the model at birth, 9 months, and 14 months (36.0 cm, 44.7 cm, and 46.9 cm respectively) align well with our equation and demonstrate a consistent trend that can be useful for estimating head circumferences in infants.
Therefore, we conclude the answers:
10a [tex]\( \boxed{36.0 \text{ cm}} \)[/tex]
10b [tex]\( \boxed{44.7 \text{ cm}} \)[/tex]
10c [tex]\( \boxed{46.9 \text{ cm}} \)[/tex]
### a. Head Circumference at Birth
Given [tex]\( x = 0 \)[/tex]:
1. Substitute [tex]\( x \)[/tex] into the equation:
[tex]\[ y = 2.9 \sqrt{0} + 36 \][/tex]
2. Simplify the equation:
[tex]\[ y = 2.9 \times 0 + 36 = 36 \][/tex]
3. Therefore, the head circumference at birth ([tex]\( x = 0 \)[/tex]) is:
[tex]\[ \boxed{36.0 \text{ cm}} \][/tex]
### b. Head Circumference at 9 Months
Given [tex]\( x = 9 \)[/tex]:
1. Substitute [tex]\( x \)[/tex] into the equation:
[tex]\[ y = 2.9 \sqrt{9} + 36 \][/tex]
2. Simplify the equation:
[tex]\[ y = 2.9 \times 3 + 36 = 8.7 + 36 = 44.7 \][/tex]
3. Therefore, the head circumference at 9 months ([tex]\( x = 9 \)[/tex]) is:
[tex]\[ \boxed{44.7 \text{ cm}} \][/tex]
### c. Head Circumference at 14 Months
Given [tex]\( x = 14 \)[/tex]:
1. Substitute [tex]\( x \)[/tex] into the equation:
[tex]\[ y = 2.9 \sqrt{14} + 36 \][/tex]
2. Simplify the equation:
[tex]\[ y \approx 2.9 \times 3.74 + 36 \approx 10.846 + 36 = 46.9 \][/tex]
3. Therefore, the head circumference at 14 months ([tex]\( x = 14 \)[/tex]) is:
[tex]\[ \boxed{46.9 \text{ cm}} \][/tex]
### d. Validating the Model
To compare the model with empirical data on your graphing calculator:
1. Plot the equation [tex]\( y = 2.9 \sqrt{x} + 36 \)[/tex] on a graphing calculator.
2. Locate the points for [tex]\( x = 0, 9, \)[/tex] and [tex]\( 14 \)[/tex] months on the graph.
3. Verify if the calculated values from this model match the points on the graph.
From the calculations:
- At [tex]\( x = 0 \)[/tex], the graph should show [tex]\( (0, 36) \)[/tex].
- At [tex]\( x = 9 \)[/tex], the graph should show [tex]\( (9, 44.7) \)[/tex].
- At [tex]\( x = 14 \)[/tex], the graph should show [tex]\( (14, 46.9) \)[/tex].
By comparing the points on the graph with the model-derived values, you can determine if the model provides an accurate representation of the head circumference over time. If the points match closely, then this is a good model. Given that models in science and mathematics often have small deviations from actual measurements, slight differences are expected. However, if the model-derived values are close to the plotted points, then this model is likely reliable.
#### Conclusion
The values obtained from the model at birth, 9 months, and 14 months (36.0 cm, 44.7 cm, and 46.9 cm respectively) align well with our equation and demonstrate a consistent trend that can be useful for estimating head circumferences in infants.
Therefore, we conclude the answers:
10a [tex]\( \boxed{36.0 \text{ cm}} \)[/tex]
10b [tex]\( \boxed{44.7 \text{ cm}} \)[/tex]
10c [tex]\( \boxed{46.9 \text{ cm}} \)[/tex]
