Question

Question #00a87

Answer 1

The length of the fence the dog can travel is `sqrt(128)` or `8sqrt(2)`

Explanation:

Since the dog has 8 feet of chain to travel, and the line will never reach a distance longer than 8, we can assume the dog can reach all the fence distance in the positive quadrant 1. However, here is the calculation that proves it:

Find the equation of the circle the dog can travel:

`x^2 + y^2 = 64`
Substitute in our equation for y.
`x^2 + (8-x)^2 = 64`
`x^2 + 64 - 16x + x^2 = 64`
`2x^2 - 16x = 0`
Factor and solve for x
`2x( x-8) = 0`
`2x = 0`
`x = 0`
`x-8 = 0`
`x = 8`

So we know that the dog can travel the fence from the interval 0 to 8. Now we plug these x values into our equation to determine the y values, and plug the y values into our distance formula.

`y = 8 - 0 = 8`
`y = 8 - 8 = 0`
The points are (0,8) and (8,0)
`"distance" = sqrt((y_1 - y_2)^2 + (x_1 - x_2)^2)`
`"distance" = sqrt(8^2 + (-8)^2)`
`"distance" = sqrt(128)`

This is our answer, we can reduce it to `8sqrt(2)` though. Whichever works best for you.

Hope this helped!