Use the tangent ratio twice to find how far the boat is from both towers.
≈ 2.7 feet
Practice makes perfect
We are given a figure that shows a boat passing between two towers. We want to find the distance between the boat and the middle of the channel. Let's first take a look at the given figure.
As we can see, we have two right triangles. To find the middle of the channel, we will calculate the total length of the channel and then divide it by 2. To do so we need to use the missing side lengths of the right triangles. Let's consider the triangle at the left hand side of the figure.
Since we have an acute angle that is 30^(∘) and the measure of its opposite side, we can use the tangent ratio to find its adjacent side.
tan(θ) = Opposite side/Adjacent side
Let's substitute θ= 30 and opposite side= 10 to find the length of the adjacent side x.
The length of the adjacent side is about 17.3feet. Now let's continue with the triangle on the right-hand side of the figure.
Since we have an acute angle of 40^(∘) and the measure of its opposite side is 10 feet, we will use the tangent ratio one more time. Let's substitute these values into the tangent ratio and then calculate the value of y.
With the x and y values we found how far the boat is from both towers. We can now calculate the total length of the channel. Let's add them up!
Length of the Channel: 17.3+ 11.9 ≈ 29.2
Great! Now we will find the middle of the channel dividing this measure by 2.
Middle of the Channel: 29.2/2 ≈ 14.6
To find the distance between the middle of the channel and the boat, let's see the lengths that we have found on the figure.
As we can see from the figure, we can find the distance between the middle of the channel and the boat in two ways.
( 17.3- 14.6) &⇔ ( 14.6- 11.9)
2.7 &⇔ 2.7
Finally, we can conclude that the boat needs to move approximately 2.7 feet to left to be in the middle of the channel.
