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One side of a triangle is increasing at a rate of 8 cm/s and a second side is decreasing at a rate of 3 cm/s. if the area of the triangle remains constant, at what rate does the angle between the sides change when the first side is 24 cm long, the second side is 35 cm, and the angle is π/6?

Respuesta :

sqdancefan
A = (1/2)ab*sin(θ)
A' = (a'*b*sin(θ) +a*b'*sin(θ) +a*b*cos(θ)*θ')/2
0 = (8 cm/s)*(35 cm)*sin(π/6) +(24 cm)*(-3 cm/s)*sin(π/6) +(35 cm)*(24 cm)*cos(π/6)*θ'
0 = 140 cm^2/s -36 cm^2/s +420√3 cm^2*θ'
θ' = -104/(420√3) . . . . rad/s
.. = -26(√3)/315 . . . . . . rad/s
.. ≈ -8.19°/s

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The figure offers a sort of sanity check on the numbers.
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