Question

You are setting up a zip line in your yard. You map out your yard in a coordinate plane. An equation of the line representing the zip line is y=−4/5x+5. There is a tree in your yard at the point (4, 10). Each unit in the coordinate plane represents 1 foot. Approximately how far away is the tree from the zip line? Round your answer to the nearest tenth.

Answer 1

The tree is approximately 6.4 feet away from the zip line.

Explanation:

Equation of the line representing the zip line is
`y=-(4)/(5)x+5`

First we need to convert the above equation into
`ax+by+c=0` form. So........

`y=-(4)/(5)x+5\\ \\ 5y=5(-(4)/(5)x+5)\\ \\ 5y=-4x+25\\ \\ 4x+5y-25=0`

Thus,
`a=4, b=5` and
`c=-25`

The formula for distance
`(d)` from a point
`(x_(0), y_(0))` to the line
`ax+by+c=0` is.........

`d=(|a(x_(0))+b(y_(0))+c|)/(√(a^2+b^2))`

Given that, there is a tree in your yard at the point (4, 10). So here,
`x_(0)=4` and
`y_(0)=10`

Thus, the distance will be:
`d=(|4(4)+5(10)-25|)/(√(4^2+5^2))= (|16+50-25|)/(√(16+25))= (41)/(√(41))=√(41)= 6.403... \approx 6.4` (Rounding to the nearest tenth)

So, the tree is approximately 6.4 feet away from the zip line.