A child is flying a kite. If the kite is 125 feet above the child's hand level and the wind is blowing it on a horizontal course at 7 feet per second, the child is letting out cord at _____ feet per second when 245 feet of cord are out. Assume that the cord remains straight from hand to kite.

Kite string angle relative to horizontal = sin^-1(125/245) = 30.7 degrees.

String horizontal location = x
dx/dt = 7 ft/s
L = sqrt (x^2 + y^2)
(y is constant at 125 ft)

You want dL/dt = (dL/dx)*(dx/dt)
dL/dx = x/sqrt(x^2 + y^2)]

Take it from there.

To figure out the speed at which the child is letting out cord, we can use the concept of similar triangles.

Let's create a diagram to understand the situation better:

```
Kite
(125 ft)
/|\
/ | \
/ | \
/ | \
/ | \
/ | \
/ | \
/ | \
Child | Ground
(hand) |
```

We can consider two similar triangles here: the first triangle formed by the child's hand, the kite, and the ground, and the second triangle formed by the child's hand, the kite, and the length of the cord. Since the triangles are similar, their corresponding sides are proportional.

Let's assign variables to the distances involved:
- The vertical distance from the kite to the ground is 125 feet (let's call it 'h').
- The horizontal distance traveled by the kite is unknown (let's call it 'x').
- The distance of the cord let out by the child is 245 feet (let's call it 'c').

From the given information, we can determine the value of 'x' using the concept of similar triangles:

```
x / c = h / (h + c) <-- Similar triangles proportionality
```

Substituting the known values:

```
x / 245 = 125 / (125 + 245)
x / 245 = 125 / 370
x = 245 * (125 / 370)
x ≈ 82.8378 feet
```

Now that we know the horizontal distance 'x' traveled by the kite, we can find the time it takes to travel this distance by using the formula:

```
time = distance / speed
```

Given that the horizontal speed of the kite is 7 feet per second, we can calculate the time:

```
time = x / 7
time ≈ 82.8378 / 7
time ≈ 11.8341 seconds
```

Finally, to determine the speed at which the child is letting out the cord when 245 feet are out, we can divide the distance let out by the time taken:

```
speed = c / time
speed = 245 / 11.8341
speed ≈ 20.7255 feet per second
```

Therefore, when 245 feet of cord are out, the child is letting it out at approximately 20.7255 feet per second.