Two sides of a triangle have lengths 12 m and 14 m. The angle between them is increasing at a rate of 2°/min. How fast is the length of the

Question

Two sides of a triangle have lengths 12 m and 14 m. The angle between them is increasing at a rate of 2°/min. How fast is the length of the third side increasing when the angle between the sides of fixed length is 60°? (Round your answer to three decimal places.) webassig n

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Ayla 3 months 2021-10-13T22:20:52+00:00 1 Answer 0 views 0

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    2021-10-13T22:22:21+00:00

    Answer:

    1.692 m/min

    Step-by-step explanation:

    Let \theta be the angle between the two sides and x be the length of the third side. By cosine rule,

    x^2 = 12^2+14^2-2\times12\times14\cos\theta = 340 - 336\cos\theta

    x= \sqrt{340 - 336\cos\theta}

    We differentiate x with respect to \theta by applying chain rule.

    \dfrac{dx}{d\theta} = \dfrac{336\sin\theta}{2\sqrt{340 - 336\cos\theta}} = \dfrac{168\sin\theta}{\sqrt{340 - 336\cos\theta}}

    Rate of change of \theta is 2

    \dfrac{\theta}{dt} = 2

    Rate of change of x is

    \dfrac{dx}{dt} = \dfrac{dx}{d\theta}\times\dfrac{d\theta}{dt}

    \dfrac{dx}{dt} = \dfrac{168\sin\theta}{\sqrt{340 - 336\cos\theta}} \times2=\dfrac{336\sin\theta}{\sqrt{340 - 336\cos\theta}}

    At 60°,

    \dfrac{dx}{dt} = \dfrac{336\sin60}{\sqrt{340 - 336\cos60}} = 1.692 \text{ m/min}

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