Anonymous
Anonymous asked in Science & MathematicsMathematics · 1 month ago

related rates question?

A particle is moving along the graph of y = ∛x. When x = 8, the y-component of the position of the particle is increasing at the rate of 1 centimeter per second. How fast is the angle of inclination θ changing at this moment?

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  • 1 month ago

    y = cbrt(x) = x^(1/3)

    x = y^3

    tan(θ) = y/x = y/(y^3) = 1/y^2θ = arctan(1/y^2)dθ/dy = -2y / (y^4 + 1)

    When x = 8, y = cbrt(8) = 2:

    dθ/dy = -2*2 / (2^4 + 1) = -4 / (16 + 1) = -4/17

    dy/dt = 1

    dθ/dt = dθ/dy * dy/dt = (-4/17) * 1 = -4/17 radians per second

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