Respuesta :

Answer:

[tex]\frac{d\theta}{dt}=-\frac{2}{5}[/tex] at [tex]x=12[/tex]

Step-by-step explanation:

[tex]\frac{dx}{dt}=2[/tex]

[tex]\frac{d\theta}{dt}=?[/tex]

[tex]x=12[/tex]

[tex]cos(\theta)=\frac{adjacent}{hypotenuse}[/tex]

[tex]cos(\theta)=\frac{x}{13}[/tex]

[tex]\frac{d}{dt}cos(\theta)=\frac{d}{dt}\frac{x}{13}[/tex]

[tex]-sin(\theta)\frac{d\theta}{dt}=\frac{1}{13}\frac{dx}{dt}[/tex]

[tex]-sin(\theta)\frac{d\theta}{dt}=\frac{1}{13}(2)[/tex]

[tex]-sin(\theta)\frac{d\theta}{dt}=\frac{2}{13}[/tex]

[tex]\frac{d\theta}{dt}=\frac{2}{-13sin(\theta)}[/tex]

[tex]cos(\theta)=\frac{x}{13}[/tex]

[tex]cos(\theta)=\frac{12}{13}[/tex]

[tex]\theta=cos^{-1}(\frac{12}{13})[/tex]

[tex]\frac{d\theta}{dt}=\frac{2}{-13sin(\theta)}[/tex]

[tex]\frac{d\theta}{dt}=\frac{2}{-13sin(cos^{-1}(\frac{12}{13}))}[/tex]

[tex]\frac{d\theta}{dt}=\frac{2}{-13(\frac{5}{13})}[/tex]

[tex]\frac{d\theta}{dt}=\frac{2}{-5}[/tex]

[tex]\frac{d\theta}{dt}=-\frac{2}{5}[/tex]

The value of θ is given by the inverse sine function, from which the rate of

change of θ with respect to x can be derived.

  • The rate of change of θ at the instant when x = 12 units is -0.4 rad/s

Reasons:

[tex]\displaystyle \frac{dx}{dt} = \mathbf{2 \ units \ per \ second}[/tex]

[tex]\displaystyle cos(\theta) = \frac{x}{13}[/tex]

[tex]\displaystyle \theta = arccos \left(\frac{x}{13}\right)[/tex]

[tex]\displaystyle \frac{d}{dx} \theta = \frac{d\left(arccos \left(\frac{x}{13}\right)\right)}{dx} = \mathbf{\frac{\sqrt{169-x^2} }{x^2-169}}[/tex]

Using chain rule of differentiation, we have;

[tex]\displaystyle \frac{d\theta}{dt} = \mathbf{ \frac{d\theta}{dx} \times \frac{dx}{dt}}[/tex]

Therefore;

[tex]\displaystyle \frac{d\theta}{dt} =\frac{\sqrt{169-x^2} }{x^2-169}\times \frac{dx}{dt} = \mathbf{\frac{\sqrt{169-x^2} }{x^2-169}\times 2}[/tex]

When x = 12, we get;

[tex]\displaystyle \frac{d\theta}{dt} =\frac{\sqrt{169-12^2} }{12^2-169}\times 2 = -\frac{2}{5} = -0.4[/tex]

The rate of change of the angle, θ, with time at the instant when x = 12 is -0.4 rad/s

Learn more about rate of change here:

https://brainly.com/question/24686935

Otras preguntas