We are asked to estimate the distance between the Empire State Building and the Statue of Liberty. First, let's draw a diagram representing the situation. We are focusing on three objects: the Empire State Building, the Chrysler Building, and the Statue of Liberty. These objects form a triangle.
We know that m∠ B= 145^(∘), b= 5.6 miles, and c= 0.6 miles.
The distance between the Empire State Building and the Statue of Liberty is represented by d. To find the distance, we will follow three steps.
For any triangle the ratio between the length of the side and the sine value of its opposite angle is constant.
Now we will apply this law to △ BCD.
5.6/sin 145^(∘)=0.6/sinC=d/sinD
Since we want to find m∠ C, we will use the above expression to write an equation involving sinC.
5.6/sin145^(∘)=0.6/sinC
We will rewrite the obtained equation so that sinC is isolated on the left-hand side.
To find the measure of ∠ C, we will use the inverse sine ratio.
sinC=0.6sin145^(∘)/5.6
⇕
m∠ C=sin^(- 1)(0.6sin145^(∘)/5.6)
Let's use a calculator to approximate m∠ C to the nearest tenth of a degree.
m∠ C=sin^(- 1)(0.6sin145^(∘)/5.6)≈3.5^(∘)
The measure of ∠ C is about 3.5^(∘). Now we will use the Triangle Sum Theorem to find m∠ D.
m∠ D≈ 180^(∘)- 145^(∘)-3.5^(∘)=31.5^(∘)
The measure of ∠ D is about 31.5^(∘). Finally we can write an equation that can be solved for d using the Law of Sines.
5.6/sin 145^(∘)≈d/sin31.5^(∘) ⇔ d ≈5.6sin31.5^(∘)/sin145^(∘)
Let's use a calculator to estimate the value of d.
d≈5.6sin31.5^(∘)/sin145^(∘)≈ 5.1
The distance d between the Empire State Building and the Statue of Liberty is about 5.1 miles.
