After several days of camping and long-distance bicycle
riding, you decide to ride straight home riding
at a constant rate.
After 2 hours of riding, you are 10 kilometers from home.
And after 4 hours of riding, you are 5 kilometers from home.
Once you begin to ride home, in how many hours
will you finish your bike ride and arrive at home?
This last sentence is a little bit redundant.
It really should say, what was your total length
of your bike ride home?
But let's just think about the data that they gave us.
So let's just think about on one column,
let's put how long you've been riding.
So time.
Time riding.
And then in this column let's put the distance from home.
So they tell us that after 2 hours,
and the time is in hours, after 2 hours
we are 10 kilometers from home.
And after 4 hours, we are 5 kilometers from home.
And we are going home at a constant rate.
So let's think about what this rate is.
When our time increased by 2 hours,
our distance decreased by 5.
So if our time increased by another 2 hours,
so now 6 hours have gone by, then our distance
will have decreased by 5 again, and we
are 0 kilometers from home.
So we've answered our question.
Once you begin to ride home, in how many hours
will you finish your bike ride and arrive home?
Well, it'll take you a total of 6 hours.
Now, I'm going to ask ourselves another question
that they didn't ask us here.
Which is, how far away was I when I started riding home?
Well, we could back things up a little bit.
Actually, let me copy and paste all of this.
And move it down.
So let me cut, let me paste this all down a little bit,
so we have some extra space.
And let me draw a little white line here.
And then finally, let me erase this business right over here.
And so we could think of it the other way.
Where were we at time 0?
So time 0, that was 2 hours before this point
right over here.
So here we are subtracting 2 hours.
So instead of subtracting 5, we will add 5.
They're going to have opposite signs.
So we were 15 kilometers away.
Another way that you could think about this
is we started 15 kilometers away, 2 hours goes by
and our distance has decreased 5 kilometers.
So let's actually graph this just
to visualize things in graphical form.
So let's say that this is our time axis,
actually our distance axis.
Distance in this case is a function of time,
so the convention is to plot that on the vertical axis.
And then let's put time on the horizontal axis.
So that is time right over there.
So when time equals 0 our distance is 15 kilometers.
This is 5, 10, 15.
So when time is equal to 0, distance is 15 kilometers.
That's this point right over here.
And then when time is equal to 6,
our distance is 0 kilometers.
So it's not perfectly-- so let's say
that this is 1 hour, 2 hours, 3 hours, 4 hours, 5 hours, and 6
hours, obviously, it should be drawn a little bit neater
than that.
Let's do it this way.
Let's call this 2, 4, and 6 hours.
Obviously, not the same scales for
the vertical and horizontal axes.
But this last point right over here when time is 6 hours,
our distance from home is 0 kilometers.
So that's that point there.
And we're traveling at a constant rate.
So our line will look like this.
Now one question you might ask yourselves, well,
can we quantify this rate, or how would
this rate relate to the inclination of this line?
Well remember, the inclination of this line
is defined as slope, and that's the rate
of change of our vertical axis to our horizontal axis.
So the slope of this line is equal to the change
in our vertical axis.
This triangle is the Greek letter delta,
shorthand for change in.
Change in our vertical axis with respect
to a change in our horizontal axis.
So in our horizontal axis, we already
see when we have a change of 2 in our horizontal axis,
and we saw that with the first two points,
when we went from 2 comma 10, that
was this point right over here, to 4 comma
5, what was our change in time?
Well, our change in time was we went from 2 to 4.
So I could do it right over here.
So our change in time was equal to 2.
And what was our change in distance?
Well our change in distance, we went from 10 to 5.
So our change in distance was equal to negative 5.
It went down when our time increased.
So it's negative 5.
So the slope of this line is negative 5/2.
We started at time 0 at 15 kilometers,
so actually we could express the equation for this line.
We could write that distance-- and we could even
write it as distance as a function of time like that,
or we could just write distance-- is equal to,
we started at 15 kilometers, and then
we decreased our distance from there as time increases.
So it's 15 plus negative 5/2 times time,
or we could say minus 5/2 times time.
So we didn't have to do all of this.
We could have just reasoned through it with the table
that we did really in the first minute,
but it is nice to just think about this
in all the different ways.