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

increasing transnational kinetic energy. because the transnational kinetic energy formula consists of only two variables, mass and velocity, increasing one of those properties is the only way to increase an object's transnational kinetic energy. increases to mass and velocity, however, do not have the same impact.

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