Answer:
[tex]\theta = 4.8^\circ[/tex]
It is possible
Step-by-step explanation:
Given
See attachment for the construction
Solving (a): The minimum angle
This is calculated as thus:
[tex]tan(\theta) = \frac{Opposite}{Adjacent}[/tex]
[tex]tan(\theta) = \frac{15}{180}[/tex]
[tex]tan(\theta) = 0.0833[/tex]
Calculate [tex]\theta[/tex]
[tex]\theta = tan^{-1}(0.0833)[/tex]
[tex]\theta = 4.761[/tex]
[tex]\theta = 4.8^\circ[/tex] --- approximated
Solving (b):
First; calculate the height of the building using the following equivalent ratio.
[tex]h : 420 = 15 : 180[/tex]
Where h is the height of the building
[tex]\frac{h }{ 420 }= \frac{15 }{ 180}[/tex]
[tex]h= \frac{15 }{ 180}*420[/tex]
[tex]h= \frac{15*420 }{ 180}[/tex]
[tex]h= \frac{6300}{ 180}[/tex]
[tex]h= 35[/tex]
This means that it is possible that the builder builds a 32 feet high building
From the picture attached,
Angle of depression = Angle of elevation (θ) [Alternate angles]
Apply tangent rule in ΔCDP,
[tex]\text{tan}\theta=\frac{\text{Opposite side}}{\text{Adjacent side}}[/tex]
[tex]\text{tan}\theta=\frac{CD}{DP}[/tex]
[tex]=\frac{15}{180}[/tex]
[tex]=\frac{1}{12}[/tex]
[tex]\theta=\text{tan}^{-1}(0.08333)[/tex]
[tex]\theta=4.76^\circ[/tex]
Use this property to get the height of the new building.
Since, ΔABP and ΔCDP are the similar triangles, their corresponding sides will be proportional.
[tex]\frac{AB}{BP}=\frac{CD}{DP}[/tex]
[tex]\frac{AB}{420}=\frac{15}{180}[/tex]
AB = [tex]\frac{420\times 15}{180}[/tex]
= 35 feet
But the construction for a new building has a maximum height restriction of 32 feet.
And the height of the building is more than the restricted height.
Therefore, it is not possible for the builder to have a rooftop patio with a view of the ocean.
Learn more about the similar triangles here,
https://brainly.com/question/10043253?referrer=searchResults
