Problema Solution
Once Kate's kite reaches a height of 50 feet (above her hands), it rises no higher but drifts due east in a wind blowing at the rate of 5 ft/s. How fast is the string running through Kate's hands at the moment that she has released 120 ft of string?
Which sides are changing? Assign variables to them.
Which rate is given?
Which rate do you have to find?
Write an equation connecting the variables.
Differentiate.
Solve for the unknown rate.
Answer provided by our tutors
the angle is changing per the tangent function . If the opposite side from the angle is changing at 5fps, this is the derivative of the tangent function, which is the sec^2 function. If Kate has released (another) 120 feet the hypotenuse is 120 + 50, or 170 feet, and the angle is arcos(50/170) = 72.9 degrees, The derivative of the tan function is the sec^2 function, which is 1/(cos^2). Cos^2(72.9) = 0.0864. And 1/(0.0864) = 11.57 feet per second.