Question

find the smallest angle of △bcd.

Explanation:

Step1: Recall angle - side relationship

In a triangle, the smallest angle is opposite the shortest side. The shortest side of $\triangle BCD$ is $BD = 45$ ft. The angle opposite side $BD$ is $\angle C$.

Step2: Use the Law of Cosines

The Law of Cosines formula is $c^{2}=a^{2}+b^{2}-2ab\cos C$, where $a = 59$, $b = 63$, and $c = 45$. So, $45^{2}=59^{2}+63^{2}-2\times59\times63\times\cos C$.
First, calculate the squares:
$2025 = 3481+3969 - 7434\cos C$.
$2025=7450 - 7434\cos C$.
Then, solve for $\cos C$:
$7434\cos C=7450 - 2025$.
$7434\cos C = 5425$.
$\cos C=\frac{5425}{7434}\approx0.73$.

Step3: Find the angle

$C=\cos^{-1}(0.73)\approx43.1^{\circ}$.

Answer:

Approximately $43.1^{\circ}$