Question

1. navigation the paz verde, a whalewatch boat, is located at point p, and l is the nearest point on the baja california shore. point q is located 4.25 mi down the shoreline from l and \\(\overline{pl} \perp \overline{lq}\\). determine the distance that the paz verde is from the shore if \\(\angle pql = 35^\circ\\).

Explanation:

Step1: Identify the triangle type

We have a right triangle \( \triangle PLQ \) with \( \angle PLQ = 90^\circ \), \( LQ = 4.25 \) mi, and \( \angle PQL = 35^\circ \). We need to find \( PL \), the distance from the boat to the shore.

Step2: Use tangent function

In a right triangle, \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \). Here, \( \theta = 35^\circ \), opposite side to \( \theta \) is \( PL \), and adjacent side is \( LQ \). So, \( \tan(35^\circ)=\frac{PL}{LQ} \).

Step3: Solve for \( PL \)

Substitute \( LQ = 4.25 \) mi into the formula: \( PL = LQ \times \tan(35^\circ) \). Calculate \( \tan(35^\circ) \approx 0.7002 \). Then \( PL = 4.25 \times 0.7002 \approx 2.976 \) mi.

Answer:

The distance from the Paz Verde to the shore is approximately \( 2.98 \) miles (or more precisely \( \approx 2.98 \) mi, depending on the precision of \( \tan(35^\circ) \) used). If we use more precise value of \( \tan(35^\circ) \approx 0.7002075382 \), then \( 4.25\times0.7002075382\approx 2.976 \) mi, which can be rounded to \( 2.98 \) mi or \( 3.0 \) mi (depending on significant figures).