Respuesta :
Answer:
[tex]\dfrac{d\theta}{dx}=-0.001633\ rad/ft[/tex]
Explanation:
It is given that,
Height of the building, h = 245 ft
The base of the building and shadow is x.
Let [tex]\theta[/tex] is the angle of elevation that is formed by lines from the top and bottom of the building to the tip of the shadow. Using the trigonometric ratio as :
[tex]tan\ \theta=\dfrac{245}{x}[/tex]
[tex]\theta=tan^{-1}(\dfrac{245}{x})[/tex]
[tex]\dfrac{d\theta}{dx}=\dfrac{1}{1+(245/x^2)}(\dfrac{-245}{x^2})[/tex]
[tex]\dfrac{d\theta}{dx}=\dfrac{-245}{x^2+60025}[/tex]
When x = 300 feet
[tex]\dfrac{d\theta}{dx}=\dfrac{-245}{(300)^2+60025}[/tex]
[tex]\dfrac{d\theta}{dx}=-0.001633\ rad/ft[/tex]
So, the rate of change of the angle of elevation is -0.001633 rad/ft. Hence, this is the required solution.
The angle of elevation decreases as the length of the shadow which forms the angle increases
The rate of change of the angle of elevation, when x = 300 feet, is approximately -0.00163 radians per foot
Reasons:
Given parameter;
Height of the building = 245 ft.
Lengths of the shadow cast by the building = x
Angle of elevation of the top of the building to the tip of the shadow = θ
Required:
The rate of change of the angle of elevation, [tex]\mathbf{\dfrac{d \theta}{dx}}[/tex], when, x = 300 feet
Solution:
[tex]tan(\theta) = \dfrac{245}{x}[/tex]
[tex]\theta = arctan\left( \dfrac{245}{x} \right)[/tex]
Using chain rule of differentiation, we get;
[tex]Differentiation \ of \ arctan(x), \dfrac{d}{dx}(arctan(x)) = \dfrac{1}{x^2 + 1}[/tex]
[tex]\dfrac{d \theta}{dx} = \dfrac{d}{dx} \left(arctan\left( \dfrac{245}{x} \right) \right) =\left(\dfrac{1}{\left(\dfrac{245}{x} \right)^2+1}\right) \times \left( \dfrac{-245}{x^2} \right) = -\dfrac{245}{60025 + x^2}[/tex]
Therefore;
[tex]\dfrac{d \theta}{dx} = -\dfrac{245}{60025 + x^2}[/tex]
When x = 300 feet, we get;
[tex]\dfrac{d \theta}{dx} = -\dfrac{245}{60025 + 300^2} \approx -1.63306 \times 10^{-3}[/tex]
Therefore, when x = 300 feet, [tex]\dfrac{d \theta}{dx}[/tex] ≈ -0.00163 radians/ft.
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