Answer :
The horizontal distance and kite height serve as examples of rates.
At a rate of 0.36 radians per second, the angle is dropping.
These are the specified parameters:
Height = y = 100 ft
Speeds = [tex]\frac{dx}{dt}[/tex]
= 9 fts ⁻¹
Length = 200 ft
Determine the angle utilizing the following sine ratio:
sin (θ) = [tex]\frac{100}{200}[/tex]
sin (θ) = [tex]\frac{1}{2}[/tex]
The horizontal displacement (x) is computed using the tangent ratio shown below:
tan (θ) = [tex]\frac{100}{x}[/tex]
Take the opposite of both sides:
cot (θ) = [tex]\frac{x}{100}[/tex]
cot (θ) = [tex]\frac{1}{100} x[/tex]
Differentiate each side in terms of time (t):
₋csc² (θ) · [tex]\frac{d0}{dt}[/tex] = [tex]\frac{1}{100} . 9[/tex]
₋csc² (θ) · [tex]\frac{d0}{dt}[/tex] = [tex]\frac{9}{100}[/tex]
Recall here which:
sin (θ) = [tex]\frac{1}{2}[/tex]
Take the inverse of both sides:
csc (θ) = 2
Square all these sides:
csc² (θ) = 4
Substitute this one:
csc² (θ) = 4 in ₋csc² (θ) · [tex]\frac{d0}{dt}[/tex] = [tex]\frac{9}{100}[/tex]
- 4 · [tex]\frac{d0}{dt}[/tex] = [tex]\frac{9}{100}[/tex]
Divided both sides by -4
[tex]\frac{d0}{dt}[/tex] = [tex]-\frac{36}{100}[/tex]
[tex]\frac{d0}{dt}[/tex] = [tex]\frac{9}{100} = -0.36[/tex]
The angle is therefore decreasing at a rate of 0.36 radian per second.
Fact
In Geometry, an angle is a figure created by two rays or lines that share a common endpoint. "angle" comes from the Latin word "angulus," which means "corner." The two rays that make up an angle are known as the sides, and the common terminal is known as the vertex.
Learn more about other angles calculation here:
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