so obviously there is a right angled triangle here, so the formula we will be using is Pythagoras' formula.
notice that x is increasing at the same rate y is decreasing, so dx/dt = -dy/dt
also notice that when the weight is four feet from the ground, y=4, x = 26 (since its a 30 foot rope, they have to add up to 30) and by pythagoras' formula, z = 25.69
we are told the the car moves 5 feet/s so dz/dt is increasing at 5 ft/s
we are looking for dy/dt when y = 4, here goes:
x^2 = y^2 + z^2
=> 2x dx/dt = 2y dy/dt + 2z dz/dt
=> 2x (-dy/dt) = 2y dy/dt + 2z dz/dt
=> - 2x dy/dt - 2y dy/dt = 2z dz/dt
=> dy/dt (-2x - 2y) = 2z dz/dt
=> dy/dt = [2z dz/dt]/(-2x - 2y)
now plug in all the values
=> dy/dt = [2(25.69)(5)]/(-2(26) - 2(4)) = 256.9/(-52 - 8) = 256.9/(-60)
=> dy/dt = - 4.28 ft/s
note that dy/dt is negative since the length of y according to our diagram is decreasing.
so the rate is just 4.28 ft/s