A rancher has 300 yards of fence
with which to enclose three sides of a rectangular garden.
The fourth side is a cliff wall
and will not require fencing.
Look at our diagram here, we'll let this black side
be the side where there's a cliff
and so the rancher has to fence these three sides.
We're asked to find the dimensions of the garden
with the largest possible area.
For the purpose of this problem the width will be
the smaller dimension needing two sides.
The length will be the longer dimension needing one side.
So, we'll label the length of these two sides w.
The length of this side l.
And, because the rancher had 300 yards of fencing
we know that l plus w plus w
or l plus two w must equal 300.
Now, our goal here is to maximize the area
of the rectangular garden and the area of a rectangle
is equal to length times width.
To maximize the area, though, we want to have
the area equation in terms of one variable rather than two.
So, we'll use this equation here
which is called a constraint,
solve for l and then perform a substitution for l
in the area equation.
To solve this equation for l we'd subtract two w
on both sides so l is equal to 300 minus two w.
From here, if l is equal to 300 minus two w,
we can substitute 300 minus two w for l
in our area equation.
So, we'd have the area is equal to
the quantity 300
minus two w times w.
Now, [mumbles] the parentheses
so we'd have the area equals,
let's put the w squared term first,
so we have negative two w squared plus 300 w.
Now that we have area expressed in terms of one variable,
the variable w, we should recognize that we have a
quadratic function or a quadratic equation
and, therefore, if we were to graph this,
we would have a parabola.
And, because the coefficient of the w squared term
is negative, we'd have a parabola that opens down.
So, remember in general, when we have
a quadratic function in this form here,
if a is positive, the parabola opens up.
If a is negative, the parabola opens down.
So, again we have a parabola that looks like this,
it opens down.
And, because we're trying to maximize the area,
we need to find the coordinates of this high point,
called the vertex.
And, when our quadratic function is in this form here,
the x coordinate of the vertex is equal to negative b
divided by two a.
And, the y coordinate is equal to f of negative b
divided by two a.
Which means to find the y coordinate,
we substitute the value of negative b divided by two a
into the function for the input variable.
Course for our equation, x is equal to w
and f x is equal to a.
Looking at our specific graph, for our area equation,
we have A equals negative two w squared plus 300 w.
And, the parabola looks like this.
When graphing with software, though, we normally have to use
y and x rather than a and w.
But, our goal here is to find the coordinates
of the vertex, this point here,
where the first coordinate is going to give us
the w value or width of the rectangle
that will maximize the area of the garden.
And, the A coordinate
is going to give us what that maximum area would be.
So, as for our equation, A is equal to negative two,
b is equal to positive 300 and we don't need c
to find the vertex,
but c is equal to zero because there's no constant term.
Let's go the next slide and find the coordinates
of the vertex.
So, again the x coordinate,
or in our case, the w coordinate of the vertex
is equal to negative b divided by two a
which would be equal to negative 300
divided by two times negative two.
So, we have negative 300 divided by negative four
which equals positive 75.
So, the w coordinate of the vertex is 75.
And, now to find the A coordinate or the maximum area,
we need to substitute w equals 75 into our area equation.
So, let's say A of 75 is equal negative two
times 75 squared
plus 300 times 75.
Of course, we could also use the factored form
of our area equation which we have here,
but we'll go ahead and leave it in this form.
And, now we'll go to the calculator.
We have negative two and then in parentheses
75 squared
plus 300
times 75
which gives us 11,250.
So, the coordinates of the vertex are 75 comma 11,250.
Let's go back to our graph and just verify
these values are correct.
Notice how the w coordinate is 75
and the A coordinate does look like it's 11,250.
And going back to our first slide,
we now know the width is equal to 75 yards
to maximize the area.
We also know the largest possible area
is 11,250 square yards, but we still don't know the length.
But, we can find the length using our equation here
where l equals 300 minus two w.
If l equals 300 minus two w and w equals 75,
l is equal to 300
minus two times 75.
So, we'd have 300 minus 150
which equals 150.
So, the length is equal to 150 yards
to maximize the area.
I hope you found this helpful.