Steve hikes from the campground to the waterfall at a steady pace. After 2 hours he is 13 miles from the waterfall and after 4 hours he is 6 miles from the waterfall.

Let's find the equation of motion for Steve.

First, the things we know:

He hikes at a steady pace, so we can model this with a linear equation:

Distance = speed*time + initial position

or:

D(t) = S*t + p.

Where we make the assumption that the waterfall is our position 0m and the t = 0h is when he starts hiking.

Now we have two data points:

"After 2 hours he is 13 miles from the  waterfall"

We can represent this with the point (2h, 13mi)

"After 4 hours he is 6 miles from the waterfall."

We can represent this with the point (4h, 6mi).

The speed will be equal to the quotient between the total distance and the total time, this is:

S = (6mi - 13mi)/(4h - 2h) = -7mi/2h = -3.5 mph.

Then we can write the equation as:

D(t) = -3.5mph*t + p.

And we know that, when t = 2h, we have D(2h) = 13mi, replacing those in our equation we can find the value of p.

13mi = -3.5mph*2h + p

13mi = -7mi + p

13mi + 7mi = 20mi = p

Then the equation of motion is:

D(t) = -3.5mph*t + 20mi.

Whit this equation we can find the total time that Steeve needs to reach the waterfall.

Remember that the waterfall is at the position 0m, then he will arrive at the waterfall when:

  D(t) = 0m = -3.5mph*t + 20mi

  3.5mph*t = 20mi

                t = 20mi/3.5h = 5.7h

So he will get to the waterfall 5.7 hours after he begins hiking.

x=4

step-by-step explanation: