Chapter 8.2, Problem 50AYU

a.

To determine

Find the distance between ship and lighthouse Q.

Answer to Problem 50AYU

  $\boxed{3.21miles}$

Explanation of Solution

Given information:

The navigator of a ship at sea spots two lighthouses that she knows to be 3 miles apart along a straight seashore. She determines that the angles formed between two line-of-sight observations of the lighthouse and the line from the ship directly to shore are $15°$ and $35°$ . See the illustration.

How far is the ship from lighthouse p?

Calculation:

Draw a diagram to represent the problem.

  

Consider the triangle OPR, we know that $P+O+R=180°$ .

  $P+90°+15°=180°$

  $P+105°=180°$

  $P=75°$

The measure of angle R is the sum of the measures $15°$ and $35°$ .

  $\angle R=15°+35°=50°$

We know that $P+Q+R=180°$

  $75°+Q+50°=180°$

  $Q+125°=180°$

  $Q=55°$

Use the law of sines to find q.

  $\frac{\sin Q}{q}=\frac{\sin R}{r}$

Substitute $55°$ for Q, $50°$ for R, and 3 for r.

  $\frac{\sin 55°}{q}=\frac{\sin 50°}{3}$

Solve for q.

  $q=\frac{3\sin 55°}{\sin 50°}$

  $\approx 3.21$

The ship is about 3.21 miles from the light house P.

b.

To determine

Find the distance between ship and lighthouse Q.

Answer to Problem 50AYU

  $\boxed{3.78miles}$

Explanation of Solution

Given information:

The navigator of a ship at sea spots two lighthouses that she knows to be 3 miles apart along a straight seashore. She determines that the angles formed between two line-of-sight observations of the lighthouse and the line from the ship directly to shore are $15°$ and $35°$ . See the illustration.

How far is the ship from lighthouse Q?

Calculation:

Use the law of sines to find p.

  $\frac{\sin P}{p}=\frac{\sin R}{r}$

Substitute $75°$ for P, $50°$ for R, and 3 for r.

  $\frac{\sin 75°}{p}=\frac{\sin 50°}{3}$

Solve for P.

  $p=\frac{3\sin 75°}{\sin 50°}$

  $\approx 3.78$

  $\boxed{3.78miles}$

Hence, the ship is about $3.78miles$ from the light house Q.

c.

To determine

Find the distance of ship from shore?

Answer to Problem 50AYU

  $\boxed{3.1miles}$

Explanation of Solution

Given information:

How far is the ship from shore?

Calculation:

We know that $h=q\sin P$ .

Substitute the know values and evaluate.

  $h=3.21\sin 75°$

  $\approx 3.1$

The ship is about $3.1miles$ from the shore.