To find the length of the building's shadow, we can use the concept of similar triangles. Since both the tree and the building cast shadows in the same direction, the ratios of their heights to their shadow lengths will be equal.
Let \( h_t \) be the height of the tree, \( s_t \) be the length of the tree's shadow, \( h_b \) be the height of the building, and \( s_b \) be the length of the building's shadow.
We know:
- The height of the tree \( h_t = 5 \) feet
- The shadow of the tree \( s_t = 3 \) feet
- The height of the building \( h_b = 45 \) feet
- The length of the building's shadow \( s_b \) is what we want to find.
The ratio can be set up as follows:
\[ \frac{h_t}{s_t} = \frac{h_b}{s_b} \]
Plugging in the known values:
\[ \frac{5}{3} = \frac{45}{s_b} \]
Now, we can solve for \( s_b \). Cross-multiplying gives:
\[ 5 \cdot s_b = 3 \cdot 45 \]
Calculating \( 3 \cdot 45 \):
\[ 3 \cdot 45 = 135 \]
So we have:
\[ 5 \cdot s_b = 135 \]
Dividing both sides by 5:
\[ s_b = \frac{135}{5} = 27 \]
Thus, the length of the building's shadow is \( \boxed{27} \) feet.