Answer :
To find the height of the tree, we can use the concept of similar triangles. When an object and its shadow form a triangle, like in this scenario, the triangles formed are similar. This means that the ratios of corresponding sides are equal.
Given:
- Nicole's height: [tex]1.55[/tex] meters
- Nicole's shadow length: [tex]25.6[/tex] meters (distance from the tree to where Nicole stands)
- The tree's shadow length: [tex]30.05[/tex] meters
Let's denote:
- [tex]h_t[/tex] as the height of the tree that we need to find.
- [tex]h_n[/tex] as Nicole's height which is [tex]1.55[/tex] meters.
- [tex]s_n[/tex] as the length of Nicole's shadow which can be considered [tex]25.6[/tex] meters with respect to the base of the tree.
- [tex]s_t[/tex] as the tree's shadow length, [tex]30.05[/tex] meters.
The triangles formed by Nicole and the tree with their shadows will be similar, so we can set up the proportion:
[tex]\frac{h_t}{s_t} = \frac{h_n}{s_n}[/tex]
Substitute the known values into the equation:
[tex]\frac{h_t}{30.05} = \frac{1.55}{25.6}[/tex]
Now, solve for [tex]h_t[/tex]:
Cross-multiply to find [tex]h_t[/tex]:
[tex]h_t = \frac{1.55 \times 30.05}{25.6}[/tex]Calculate the value:
[tex]h_t = \frac{46.5775}{25.6} \approx 1.82[/tex]
Thus, the height of the tree to the nearest hundredth of a meter is approximately [tex]1.82[/tex] meters.
