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?

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