Archimedes is draining his bathtub.
Every 2 minutes that pass, 7 gallons of water are drained.
Which of the following functions can represent the number
of gallons of water in the tub, W,
as a function of the minutes that have passed, T?
So let's think about this visually.
So let's plot the number of gallons of water we have.
Let's put that in the vertical axis.
That is a function of time.
As time passes, we will lose water at a fixed rate.
So this is our time axis right over here.
So we're going to start with some initial volume of water
in the tub.
And they actually don't tell us what
our initial volume of water is.
But let's just call that capital W sub 0.
So this is just the initial volume of water.
We're doing everything in gallons,
so let's say the initial number of gallons
or the number of gallons at time equals 0.
This is T equals zero right over here.
Then they tell us it every 2 minutes that pass-- so time
is going to be given in minutes.
Time is in minutes, and the W is in gallons-- 7 gallons of water
are drained.
So let's say 2 minutes pass from time equals 0.
So this is 1 minute, and this is 2 minutes right over here.
So we have 2 minutes passing.
And how much water is going to get drained?
Well they tell 7 gallons of water are drained.
So after 2 minutes, we're going to get to W0 minus 7.
However much water we had before,
we're going to have 7 gallons less.
W0 minus 7 is where we go.
And so if we were to plot the amount of gallons
or the number of gallons we have as a function of time,
it would look something like this.
It would look like a line that goes something like that.
So one thing that we could ask ourselves
is, what is the rate of change of W with respect to T?
What is the rate of change of our vertical axis with respect
to our horizontal axis?
And that's also known as the slope of our line.
So we could write that as the rate of change--
this Delta symbol, this Greek letter, this triangle,
is just shorthand for change in-- so our change in W
for a given change in time, well we
know that when time increases by 2 minutes-- and time
is in minutes-- W decreases by 7 gallons.
Because it decreases, it's going down.
And we see that.
This is a downward sloping line right over here.
And we see it graphically right over here.
When our change in time is equal to positive 2,
our change in volume-- so this is this distance, which
is the same thing as this distance right over here--
our change in the number of gallons is equal to negative 7.
So this right over here is the slope of our line
or the rate of change of the number of gallons
with respect to time.
Now, we don't know the exact starting position
or the exact W intercept.
But we just called it W sub 0.
So we could write a general equation here.
We could say that the number of gallons
is equal to the initial gallons that we
have plus negative 7/2 times time.
Or we could just call this minus 7/2--
that's just the negative 7/2 right over there-- times time.
And this should make complete sense now.
We have our vertical axis intercept, often
called the y-intercept.
But this isn't called the y-axis anymore.
We have our vertical intercept.
And we also have the slope of our line here.
If you're familiar with slope-intercept form,
we'd normally write y is equal to mx plus b, where this
is the slope and this is the y-intercept.
We've written it the exact same way.
We've said W is equal to negative 7/2 T plus W0.
I just swapped these two things around.
And notice, this is the slope, and this is the y-intercept.
So which of these match that pattern?
Well let's see, I should start with some positive number
of gallons.
And I'm subtracting 7/2 T. So the only one
that matches this is this one right over here,
is 50 minus 7/2 T. Now even if you didn't think anything
about slopes and you were just looking at these choices here,
you would say, well look, I should
start with a certain number of gallons of water,
and then it should only decreases
as the amount of time passes.
This one wouldn't satisfy this.
At time equals 0, I have no gallons here.
That doesn't meet the constraints here.
I have a certain number of gallons
here, so this doesn't make sense.
Here at time equals 0, I have 100 gallons.
But then as time passes, that will only increase.
So this is the faucet's on here.
It's not being drained.
So that doesn't work.
Here, once again, I'm starting with a certain number
of gallons when time equals 0.
And then as time passes, it's increasing
my number of gallons.
So once again, the faucet is on right over here.
So once again, the only one where
I'm starting with a certain number of gallons
and then as time passes my number of gallons
are reducing-- because this is a negative value--
is this choice right over here.