Question

salmon
as shown in the diagram below, an island (i) is due north of a marina (m). a boat house (h) is 4.5 miles due west of the marina. from the boat house, the island is located at an angle of 54° from the marina.
determine and state, to the nearest tenth of a mile, the distance from the boat house (h) to the island (i).
determine and state, to the nearest tenth of a mile, the distance from the island (i) to the marina (m).

Explanation:

Step1: Find HI (H to I)

We use cosine: $\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here, $\theta=54^\circ$, adjacent $HM=4.5$ mi, hypotenuse $HI$.
$\cos(54^\circ)=\frac{4.5}{HI}$
$HI=\frac{4.5}{\cos(54^\circ)}$
$HI\approx\frac{4.5}{0.5878}\approx7.7$ mi

Step2: Find IM (I to M)

We use tangent: $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$. Here, opposite $IM$, adjacent $HM=4.5$ mi, $\theta=54^\circ$.
$\tan(54^\circ)=\frac{IM}{4.5}$
$IM=4.5\times\tan(54^\circ)$
$IM\approx4.5\times1.3764\approx6.2$ mi

Answer:

Distance from boat house (H) to island (I): $\approx7.7$ miles
Distance from island (I) to marina (M): $\approx6.2$ miles