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The tortoise and the hare are racing
to see who can run 100 feet in the shortest amount of time.
Their positions in feet as a function of time in seconds,
T of t and H of t respectively, are plotted below.
So T of t is the position of the tortoise as a function of time,
and it's in feet.
H of t is the position of the hare.
What is the meaning of the rate of change
of T of t on the interval t is greater than 0
and less than 20?
So T of t, they were talking about the tortoise here.
This is the function T of t right over here,
this blue line.
And it looks like it's moving at a constant rate of change.
The slope of this line is constant.
So let's see what this rate of change is.
If we start-- so we could even start
from the starting position.
When 1 second goes by, each notch here
in the horizontal axis is 1 second.
You see 1, 2, 3, 4, 5.
So every time 1 second goes by, it looks like--
and we have to be very careful here.
At first it looks like, oh, maybe he moves 1 feet.
But we have to be careful about what these notches represent.
It looks like these notches represent 5, 10, 15, 20,
25 feet.
So every time 1 second goes by, he moves forward 5 feet.
Another second goes by, he moves forward 5 feet.
So it looks like the rate of the tortoise,
so I'll say the tortoise rate, the rate
of the change of distance with respect to time
is 5 feet for every second.
So let's see which of our choices seem to match that.
And that is over the entire interval.
The interval starts here, this is 0,
and 20 is right over here.
And it looks like we have a constant rate,
we have a constant slope.
We have the same inclination of this line
the entire time all the way until t is equal to 20.
So this first choice says the tortoise walks
at 5 feet per second the whole time, and this looks right.
Our tortoise has a constant rate of 5 feet per second.
This walks might be an understatement.
5 feet per second is a pretty good speed, especially
for a tortoise.
But we are going to go with that one.
Now, let's just look at the other ones for fun
to see how true they are or see if they could have
somehow applied to this problem.
The hare stops moving after 5 seconds.
So let's see, this is time 0, this is 5 seconds.
So if we look at the hare's position after 5 seconds,
so after time equals 5 seconds, it
does indeed look like the hare's position-- the hare is just
sitting there at the 90-foot marker line
and is not moving at all.
The rate of change of position with respect
to time for the hare at that point is 0.
So this does seem true, it just doesn't apply to the question.
It's not the meaning of the rate of change
of T of t on this interval.
This is applying to H of t.
The hare runs at a speed of 18 feet per second for the first 5
seconds.
So let's see if this is true.
For the first 5 seconds, so this is 5 seconds going by,
the hare is able to cover a distance of 90 feet.
So the hare is able to go 90 feet in 5 seconds,
so for the first 5 seconds, the hare
is-- hare's rate is 90 feet.
You could say average rate of change is 90 feet in 5 seconds.
And 90 divided by 5 is indeed 18 feet per second.
So once again, this third statement is true,
it just doesn't apply to T of t.
It's not addressing the question.
Now this fourth choice is that the hare wins the race.
Now, winning the race, they have to get
to 100 feet in the shortest amount of time.
This is 100 feet right over here.
So let's see.
So it looks like only the tortoise even
gets to that point.
At 20 seconds, the tortoise gets to the 100-foot mark.
The hare is still hanging around taking a nap or something
at 90 feet.
So this is actually outright false.
The tortoise wins the race.
So we will definitely go with our first choice.