Let's walk through the steps to determine the height of the lamppost, given the information about the boy and his shadow.
1. Identify the Known Values:
- Height of the boy: [tex]\(1.8\)[/tex] meters
- Distance from the boy to the lamppost: [tex]\(1.0\)[/tex] meter
- Length of the boy's shadow: [tex]\(2.0\)[/tex] meters
2. Understand Similar Triangles:
- The boy and his shadow form one triangle.
- The lamppost and the combined distance of the boy and his shadow form another, larger, similar triangle.
3. Set Up the Proportion Using Similar Triangles:
- The ratio of the height of the boy to the total distance (from the boy to the end of his shadow) should be the same as the ratio of the height of the lamppost to the distance from the boy to the lamppost.
4. Total Distance:
- [tex]\( \text{Total distance} = \text{distance from the boy to the lamppost} + \text{length of the shadow} \)[/tex]
- Total distance = [tex]\( 1.0 + 2.0 = 3.0 \)[/tex] meters
5. Set Up the Proportion:
[tex]\[
\frac{\text{Height of the boy}}{\text{Total distance}} = \frac{\text{Height of the lamppost}}{\text{Distance from the boy to the lamppost}}
\][/tex]
[tex]\[
\frac{1.8 \text{ meters}}{3.0 \text{ meters}} = \frac{\text{Height of the lamppost}}{1.0 \text{ meter}}
\][/tex]
6. Solve for Height of the Lamppost:
[tex]\[
\text{Height of the lamppost} = 1.8 \text{ meters} \times \frac{3.0}{1.0}
\][/tex]
[tex]\[
\text{Height of the lamppost} = 1.8 \times 3.0
\][/tex]
[tex]\[
\text{Height of the lamppost} = 5.4 \text{ meters}
\][/tex]
Thus, the height of the lamppost is 5.4 meters.
