Question

kite is flying at a height of 40 ft. a child is flying the kite so that it is moving horizontally at a rateof 3 ft/sec. if the string is taut, at what rate is the string being released when the length of the releasedstring is 50 ft?

Answer 1

To find the rate at which the string is being released, we can use the concept of similar triangles and the Pythagorean theorem. We can differentiate the equation with respect to time and solve for the rate of release. Substituting the values and solving the equation will give us the desired rate.

Step-by-step explanation:

To find the rate at which the string is being released, we can use the concept of similar triangles. Let's assume that the length of the released string at any given time is 'x'. Using the Pythagorean theorem, we can find the height 'h' above the ground where the kite is flying. We have the equation x^2 + h^2 = 40^2.

Next, we can differentiate both sides of the equation with respect to time 't' to find the rate at which the string is being released. The derivative of x^2 with respect to t is 2x(dx/dt), and the derivative of h^2 with respect to t is 2h(dh/dt). Since dx/dt is the rate at which the string is being released, we can substitute it in the equation as follows: 2x(dx/dt) + 2h(dh/dt) = 0.

Now, let's find the value of 'x' when the length of the released string is 50 ft. Using the equation x^2 + h^2 = 40^2, we have 50^2 + h^2 = 40^2. Solving for 'h', we get h = sqrt(40^2 - 50^2). Substituting this value in the equation 2x(dx/dt) + 2h(dh/dt) = 0, we can solve for dx/dt, which will give us the rate at which the string is being released.