Of course! To find the height of the lamp post, let's use the properties of similar triangles. We observe that the girl and the lamp post, along with their respective shadows, form two similar triangles. This property allows us to set up a proportion based on their heights and shadow lengths.
Here’s the detailed, step-by-step solution:
1. Given Data:
- The height of the girl: [tex]\( \(h_g = 1.6 \,\text{m}\)[/tex]
- The distance from the girl to the lamp post: [tex]\(d_g = 4.68 \,\text{m}\)[/tex]
- The length of the girl's shadow: [tex]\(s_g = 2.88 \,\text{m}\)[/tex]
2. Define the Variables:
- Let [tex]\(h_l\)[/tex] be the height of the lamp post we need to find.
- Since the girl's shadow is cast by the lamp, the total distance from the lamp post to the end of the shadow is [tex]\(d_g + s_g\)[/tex].
3. Set Up the Proportion:
The key property from the similar triangles is:
[tex]\[
\frac{\text{Height of the girl}}{\text{Length of the girl's shadow}} = \frac{\text{Height of the lamp post}}{\text{Total distance from the lamp post to the end of the shadow}}
\][/tex]
Using the given values:
[tex]\[
\frac{h_g}{s_g} = \frac{h_l}{d_g + s_g}
\][/tex]
4. Substitute the Known Values:
[tex]\[
\frac{1.6 \, \text{m}}{2.88 \, \text{m}} = \frac{h_l}{4.68 \, \text{m} + 2.88 \, \text{m}}
\][/tex]
5. Simplify the Total Distance:
[tex]\[
4.68 \, \text{m} + 2.88 \, \text{m} = 7.56 \, \text{m}
\][/tex]
6. Rewrite the Proportion:
[tex]\[
\frac{1.6 \, \text{m}}{2.88 \, \text{m}} = \frac{h_l}{7.56 \, \text{m}}
\][/tex]
7. Solve for the Height of the Lamp Post:
[tex]\[
h_l = \left(\frac{1.6 \, \text{m}}{2.88 \, \text{m}}\right) \times 7.56 \, \text{m}
\][/tex]
8. Calculate the Result:
By performing the calculation, we find:
[tex]\[
h_l = 4.2 \, \text{m}
\][/tex]
Therefore, the height of the lamp post is [tex]\(4.2\)[/tex] meters.
