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Nicole is 1.55 m tall at 2 PM. She measures the length of a tree shadow to be 30.05 m. She stands 25.6 m away from the tree. Find the height of the tree to the nearest hundredth of a meter.

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]:

  1. Cross-multiply to find [tex]h_t[/tex]:
    [tex]h_t = \frac{1.55 \times 30.05}{25.6}[/tex]

  2. 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.

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