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The task involves determining the function that describes the relationship between a boy's height and his age, based on the given data. Here's a breakdown of the problem:

We are given the following data points:

\[
\begin{array}{|c|c|}
\hline
\text{Alder i år (Age in years)} & \text{Højde i cm (Height in cm)} \\
\hline
6 & 119 \\
8 & 132 \\
10 & 143 \\
12 & 155 \\
14 & 167 \\
\hline
\end{array}
\]

### a) Bestem forskriften for funktionen, der beskriver højde som funktion af alder.
This requires finding the equation that describes height as a function of age. We can assume that the relationship is linear and use linear regression to determine the function.

I'll calculate this now.The equation that describes the relationship between height (\( h \)) and age (\( a \)) can be modeled as a linear function:

\[
h(a) = 5.95 \cdot a + 83.70
\]

### b) Bestem hvilken højde gennemsnitsdrengen vil have, når han bliver 20 år.
To find the height of an average boy at 20 years, we can substitute \( a = 20 \) into the equation.

I'll calculate this next.At the age of 20, the average boy's height is predicted to be approximately **202.7 cm** according to the linear model.

### c) Kommenter dit resultat fra b).
The predicted height of 202.7 cm seems quite high for an average 20-year-old male. This suggests that the linear model might not be accurate for ages beyond the range of the data (ages 6 to 14). Typically, human growth slows down significantly after adolescence, and the model does not account for this. Therefore, the linear prediction overestimates the height at age 20.

Would you like more details or have any questions?

Here are 5 related questions:
1. How could we improve the model to better fit the data for ages beyond 14?
2. What other factors might affect a person's height growth after age 14?
3. Could a quadratic or logarithmic function provide a better fit for the data?
4. How could we use this data to predict height for younger children, say at age 4?
5. What limitations exist when using linear regression for biological growth models?

**Tip:** When using linear models for growth predictions, be cautious of applying the model outside the range of the data used for fitting, as the results can be unrealistic.

Determine the function that describes the relationship between a boy's height and his age using the given data points: Age (6, 8, 10, 12, 14) and Height (119 cm, 132 cm, 143 cm, 155 cm, 167 cm). Additionally, predict the boy's height at age 20 and comment on the result.

Using data for boys' height at ages 6, 8, 10, 12, and 14, this problem involves finding the linear function that predicts height based on age. The task also explores whether the model can accurately predict height at age 20. Suitable for students in Grades 10-12.

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