Step 1 :First, we need to calculate the value of \(x\) when \(z = 125\) feet. We can do this by rearranging the Pythagorean theorem to solve for \(x\): \(x = \sqrt{z^2 - y^2}\).
Step 2 :Then, we can differentiate the Pythagorean theorem with respect to time \(t\) to get \(2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 2z\frac{dz}{dt}\).
Step 3 :Since \(y\) is constant (the height of the kite doesn't change), \(\frac{dy}{dt} = 0\).
Step 4 :So, the equation simplifies to \(2x\frac{dx}{dt} = 2z\frac{dz}{dt}\).
Step 5 :We can solve this equation for \(\frac{dx}{dt}\) to find the rate at which the kite is moving horizontally: \(\frac{dx}{dt} = \frac{z}{x}\frac{dz}{dt}\).
Step 6 :Finally, we can substitute the known values into this equation to find the answer. Given that \(y = 105\), \(z = 125\), and \(\frac{dz}{dt} = 2\), we find that \(x = \sqrt{z^2 - y^2} = 67.82329983125268\).
Step 7 :Substituting these values into the equation for \(\frac{dx}{dt}\), we find that \(\frac{dx}{dt} = \frac{z}{x}\frac{dz}{dt} = 3.6860489038724285\).
Step 8 :Final Answer: The rate at which the kite is moving horizontally when the string is 125 feet long is approximately \(\boxed{3.69 \, \text{feet/second}}\). This means that at that moment, the kite is moving away from the person flying it at a speed of 3.69 feet per second.