To solve for the height, [tex]\( h \)[/tex], of the tree given that Amelia is 5 feet tall and casts a 4-foot shadow, while the tree casts a 12-foot shadow, we can use the property of similar triangles.
Since the triangles formed by Amelia and her shadow, and the tree and its shadow, are similar, the ratios of their corresponding sides are equal. This means we have the following proportion:
[tex]\[ \frac{\text{height of Amelia}}{\text{length of Amelia's shadow}} = \frac{\text{height of the tree}}{\text{length of the tree's shadow}} \][/tex]
So, we can write:
[tex]\[ \frac{5}{4} = \frac{h}{12} \][/tex]
This equation can be rearranged to solve for [tex]\( h \)[/tex]:
[tex]\[ 5 \cdot 12 = 4 \cdot h \][/tex]
So, one equation is:
[tex]\[ 5 \cdot 12 = 4 \cdot h \][/tex]
Alternatively, another way to express this relationship using the similarity property is to write it as follows:
[tex]\[ \frac{4}{12} = \frac{h}{5} \][/tex]
By cross-multiplying, we can see how this is equivalent to the first equation since:
[tex]\[ 4 \cdot h = 5 \cdot 12 \][/tex]
Therefore, the two correct equations that can be used to find the height, [tex]\( h \)[/tex], of the tree are:
[tex]\[ 5 \cdot 12 = 4 \cdot h \][/tex]
[tex]\[ \frac{4}{12} = \frac{h}{5} \][/tex]
Now, let's solve one of these equations for [tex]\( h \)[/tex]:
Using [tex]\( 5 \cdot 12 = 4 \cdot h \)[/tex]:
[tex]\[ 60 = 4h \][/tex]
Solving for [tex]\( h \)[/tex]:
[tex]\[ h = \frac{60}{4} = 15 \][/tex]
Hence, the height of the tree is 15 feet.
So, the final height of the tree is 15 feet and the correctly chosen equations are:
[tex]\[ 5 \cdot 12 = 4 \cdot h \][/tex]
[tex]\[ \frac{4}{12} = \frac{h}{5} \][/tex]
Which corresponds to the given boxed answers:
[tex]\[
\begin{array}{ll}
5 h=4 \cdot 12 & \frac{4}{12}=\frac{h}{5}
\end{array}
\][/tex]
The tree height is 15 feet as well.
