Connect with a community of experts and enthusiasts on IDNLearn.com. Find accurate and detailed answers to your questions from our experienced and dedicated community members.
Sagot :
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]
Thank you for using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. IDNLearn.com provides the answers you need. Thank you for visiting, and see you next time for more valuable insights.
