a Draw a diagram representing the situation. Remember that an angle of elevation is the angle between the horizontal line of sight and the line of sight up to an object.
B
b How can the definition of a tangent help you find the height of the building?
A
a About 152.86 meters
B
b
Description: See solution. Height of the building: About 170.45 meters
Practice makes perfect
a We are asked to determine how far away is the person from the base of the cliff. Let's start by drawing a diagram representing the situation. Remember that an angle of elevation is the angle between the horizontal line of sight and the line of sight up to an object.
Let d represent the distance between the person and the base of the cliff.
To find the value of d, we will use the Law of Sines. Note that the measure of the third angle of the right triangle formed by the ground and the wall of the cliff is 27^(∘) by the Triangle Sum Theorem.
Let's apply the Law of Sines to our triangle.
d/sin 27^(∘)=300/sin 63^(∘)
Now we will solve the above equation for d.
The distance between the person and the base of the cliff is about 152.86 meters.
b We are asked to describe two methods that can be used to find the height h of the building. Then we will use one of these methods to find the height of the building. Let's consider the diagram from Part A representing the situation.
In this case, we will focus on the bigger triangle formed by the ground, the wall of the cliff, and the wall of the building. Remember that in Part A we found the distance between the person and the base of the cliff.
Method I
One way to find the height h of the building is to use the tangent of the angle of elevation to the top of the building from the ground.
tan 72^(∘)=Length of the opposite leg/Length of the adjacent leg
The length of the leg opposite the angle of elevation is equal to 300+h meters, and the length of the adjacent leg is about 152.86 meters.
tan 72^(∘)=300+h/≈ 152.86
We obtained an equation that can be solved for h.
Method II
Another way to find the height h of the building is to use the Law of Sines. Note that the measure of the third angle of the bigger right triangle is 18^(∘) by the Triangle Sum Theorem.
Now we will apply the Law of Sines to our triangle.
300+h/sin 72^(∘)=≈ 152.86/sin18^(∘)
We obtained an equation that can be solved for h.
The Height of the Building
We can use either Method I or Method II to estimate the value of h. Let's solve the equation obtained using the second method.
