Let's work through the given problem step-by-step to find the mean height of the girls in the class.
1. Total number of students and boys:
- Total students = 28
- Boys count = 17
From this, we can find the number of girls:
[tex]\[
\text{Girls count} = 28 - 17 = 11
\][/tex]
2. Mean heights:
- Mean height of boys = 1.71 meters
- Mean height of all students = 1.655 meters
We can use these mean heights to calculate the total heights.
3. Calculate the total height of all boys:
The total height of boys is given by:
[tex]\[
\text{Total height of boys} = \text{Mean height of boys} \times \text{Number of boys}
\][/tex]
Substituting the given values, we get:
[tex]\[
\text{Total height of boys} = 1.71 \times 17 = 29.07 \text{ meters}
\][/tex]
4. Calculate the total height of all students:
The total height of all students is given by:
[tex]\[
\text{Total height of all students} = \text{Mean height of all students} \times \text{Total number of students}
\][/tex]
Substituting the given values, we get:
[tex]\[
\text{Total height of all students} = 1.655 \times 28 = 46.34 \text{ meters}
\][/tex]
5. Calculate the total height of the girls:
The total height of the girls can be found by subtracting the total height of the boys from the total height of all students:
[tex]\[
\text{Total height of girls} = \text{Total height of all students} - \text{Total height of boys}
\][/tex]
Substituting the calculated values, we get:
[tex]\[
\text{Total height of girls} = 46.34 - 29.07 = 17.27 \text{ meters}
\][/tex]
6. Mean height of the girls:
Finally, the mean height of the girls is given by the total height of the girls divided by the number of girls:
[tex]\[
\text{Mean height of girls} = \frac{\text{Total height of girls}}{\text{Number of girls}}
\][/tex]
Substituting the calculated values, we get:
[tex]\[
\text{Mean height of girls} = \frac{17.27}{11} = 1.57 \text{ meters}
\][/tex]
Therefore, the mean height of the girls in the class is [tex]\(1.57\)[/tex] meters.
