Respuesta :
To answer this question it is necessary to apply the concept of the cosine of an angle and derivatives with respect to time
The solution is:
dα/dt = - 4.32 degree/s
In a right triangle the cosine of an angle is:
cos α = adjacent leg / hypothenuse
cosα = x/ L
Tacking derivatives with respect to time on both sides of equation (1)
- sinα × dα/dt = ( dx/dt × L - dL/dt × x ) / L² (1)
In this equation we know:
sinα = 2/3 dx/dt = 3 m/s L = 30 m
If L = 30 m and y = 20 m ( constant) then
Using Pithagoras´theorem
L² = x² + y²
2×L×dL/dt = 2×x×dx/dt + 0
L×dL/dt = x×dx/dt we know dx/dt = 3 m/s then
dL/dt = (20/30)×3
dL/dt = 2 m/s
And when L = 30 m x = √ (30)² - (20)² x = √900-400 x = √500
x = 22.36 m
Then by substitution in (1)
- (2/3)× dα/dt = (3×30 - 2×22.36)/ 900
- (2/3)× dα/dt = (90 - 44.72)/900
- (2/3)× dα/dt = 0.05
dα/dt = - 0.0754 ( the sign ( -) mean αangle is decreasing )
to get degrees/ second we multiply by 180 and divide by π
Then
dα/dt = - 4.32 degree/s
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