Question

A boy is flying a kite. The kite is 30 feet above his head when it gets stuck on a power wire. He begins to pull it at a rate of 5 feet per minute.

How fast is the horizontal distance changing when 50 feet of string is out?

Answer 1

The horizontal distance is changing at a rate of -25/3 feet per minute when 50 feet of string is out.

Step-by-step explanation:

To find how fast the horizontal distance is changing when 50 feet of string is out, we can use similar triangles. Let x represent the horizontal distance between the boy and the power wire, and let h represent the height of the kite above the ground. We can set up the following proportion:

(x + 50) / x = (h - 30 )/ h

Cross multiplying and simplifying, we get:

xh - 30x = hx + 50h

Re-arranging the equation to solve for x, we get:

x = (50h - 30x)/h

Now, we can differentiate both sides of the equation with respect to time:

dx/dt = (50dh/dt - 30dx/dt)/h

Since we are given that dh/dt = -5 (the height of the kite is decreasing at a rate of 5 feet per minute), and we want to find dx/dt when 50 feet of string is out, we can substitute these values and solve for dx/dt.

dx/dt = (50(-5) - 30dx/dt)/30

dx/dt = -250/30 = -25/3 feet per minute