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21 May, 03:06

7) A kite 50 ft above the ground moves horizontally at a speed of 4 ft/sec. At what rate is the length of the string increasing when 100 ft of string has been let out and the angle between the string and horizontal is decreasing at a rate of 0.3 rad/sec?

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  1. 21 May, 04:57
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    Imagine a right triangle where the legs represent the horizontal and vertical lengths of the string and the hypotenuse represents the length of the string.

    Let us assign some values:

    x = horizontal length in feet

    50 = vertical length in feet

    L = length of the string in feet

    Because we are modeling these quantities with a right triangle, we can use the Pythagorean theorem to relate them with the following equation:

    L² = x² + 50²

    We want to find an equation for the change of L over time, so first differentiate both sides with respect to time t then solve for dL/dt:

    2L (dL/dt) = 2x (dx/dt)

    dL/dt = (x/L) (dx/dt)

    First let's solve for x at the moment in time described in the problem using the Pythagorean theorem:

    L² = x² + 50²

    Given values:

    L = 100ft

    Plug in and solve for x:

    100² = x² + 50²

    x = 86.6ft

    Now let's find dL/dt. Given values:

    x = 86.6ft, L = 100ft, dx/dt = 4ft/sec

    Plug in and solve for dL/dt:

    dL/dt = (86.6/100) (4)

    dL/dt = 3.46ft/sec
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