Question

1. find x. 6 pts 100 65° 20° x

Explanation:

Step1: Find the length of the base of the left - hand right - triangle

Let the base of the left - hand right - triangle be \(a\). Using the tangent function \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), for \(\theta = 65^{\circ}\) and opposite side \(= 100\).
\(\tan65^{\circ}=\frac{100}{a}\), so \(a=\frac{100}{\tan65^{\circ}}\approx\frac{100}{2.1445}\approx46.63\)

Step2: Find the length of the entire base of the large right - triangle

Let the length of the entire base of the large right - triangle be \(b\). Using the tangent function for the large right - triangle with \(\theta = 20^{\circ}+ 65^{\circ}=85^{\circ}\) and opposite side \(=100\).
\(\tan85^{\circ}=\frac{100}{b}\), so \(b = \frac{100}{\tan85^{\circ}}\approx\frac{100}{11.4301}\approx8.75\)

Step3: Calculate \(x\)

\(x=b - a\)
\(x=\frac{100}{\tan20^{\circ}}-\frac{100}{\tan65^{\circ}}\)
Since \(\tan20^{\circ}\approx0.3640\), \(\frac{100}{\tan20^{\circ}}\approx274.73\)
\(x = 274.73-46.63\)

Answer:

\(x\approx228.1\)