The cost, in dollars, to produce x
designer dog leashes is given by C of x,
and the revenue function, in
dollars, is given by R of x.
We are first asked to determine
the profit function P of x.
Remember the product function is equal
to the revenue function,
minus the cost function
because the revenue function
gives us the money coming in,
the cost function gives
us the money going out,
and the difference, or what's
left over, is the profit.
And because we are given the
cost and revenue functions,
we can determine the profit function.
The profit function,
big P of x equals R of x
minus C of x.
This gives us the quantity
negative two x squared
plus 59 x, minus the
quantity seven x plus 10.
It is important that we have
cost function in parentheses
so that we subtract the
entire cost function.
Now we clear the parentheses
and combine like terms.
If it's helpful, we can think
of distributing a positive one
here and, because of the
subtraction, we can think
of distributing a negative one.
Distributing positive one
doesn't change anything.
We have negative two x squared, plus 59x.
When distributing negative
one, we have negative one
times seven x which
equals negative seven x
giving us minus seven x.
And then negative one times
10 is equal to negative 10,
giving us minus 10.
And we can just think of
subtracting both terms inside
which gives us minus seven x minus 10.
Now we combine like terms.
There are two x terms.
So if the profit function
big P of x equals
negative two x squared,
and then 59x minus seven x
is 52x, giving us plus 52x minus 10.
This is our profit function.
Next, we're asked to find
the number of leashes needed
to be sold to maximize the profit
and also determine the maximum profit.
What we should recognize P
of x is a quadratic function
and therefore the graph is a parabola,
and because a leading
coefficient is negative two,
the parabola opens down.
Let's take a look at a graph
of the profit function.
Again we have a parabola opening down
along the horizontal axis.
We have the number of dog leashes.
Along the vertical axis, we
have the profit in dollars.
So we should be able to recognize
that if we can determine
the ordered pair for the
vertex, this point here,
the highest point on the
graph, the first value
of the ordered pairs
is going to give us x,
the number of dog
leashes that must be sold
to maximize the profit.
And the second value of
the ordered pair is going
to be the output or function value
which will give us the maximum profit.
And because the profit
function is in general form,
or the form ax squared plus bx plus c,
we can use this formula here to determine
the ordered pair for the vertex.
So for the profit function P of x,
a is equal to negative two,
b is equal to 52,
and c is equal to negative 10.
The x coordinate of the vertex is equal
to negative b divided by two a.
This equation also gives us the equation
of the axis of symmetry.
Performing substitution, b is equal to 52,
giving us negative 52,
divided by two times a,
and a is negative two, giving
us two times negative two.
So we have x equals a negative 52,
divided by a negative four,
which is equal to positive 13.
So for the vertex,
we know the first value
of the ordered pair
or the x coordinate is 13
which means 13 dog leashes must be sold
in order to maximize the profit.
And now we need to evaluate
the profit function at 13
to determine the maximum profit.
So P of 13 is equal to negative two
times the square of 13,
plus 52 times 13, minus 10.
13 squared is equal to 169,
giving us negative two times 169,
plus 52 times 13, minus 10.
And now we multiply,
giving us negative 338,
plus 676 minus 10, and
now we add and subtract
from left to right which gives us 328.
So now we know the maximum profit is $328.
Again, the ordered pair for
the vertex is 13 comma 328.
The 13 indicates 13 dog
leashes must be sold
to maximize the profit,
and the 328 indicates the
maximum profit is $328.
Going back to our first slide,
13 dog leashes must be sold
to maximize the profit.
The maximum profit is $328.
And then finally we're asked
to find the price to charge
per leash to maximize the profit.
Let's go back to our notes for a moment.
Little p of x equals the
price-demand function in dollars,
where little p of x
represents the selling price.
And we know x represents
the quantity sold,
which means the revenue
function R of x can be expressed
as little p of x times x,
or if we want x times, little p of x.
And we need to find little
p of x in order to determine
the selling price to maximize the profit.
So we now know that the
revenue function R of x
can be expressed as little p of x times x
or x times little p of x.
Let's say x times little p of x.
And the revenue function is equal
to negative two x squared plus 59x.
So if we factor out an x, we can determine
the price demand function little p of x.
If we factor out x, we're
left with the quantity
negative two x plus 59,
where again this is x
and therefore little p of x must be equal
to negative two x plus 59.
So now that we know that
little p of x is equal
to negative two x plus
59, we can determine
the selling price to maximize the profit.
Because remember, x is equal to 13
when the profit is maximized.
So we need to find little p of 13
which is equal to negative
two times 13 plus 59
which equals negative 26 plus 59
which is equal to 33.
So now we know the price
per leash must be $33
to maximize the profit.
So going back to our
first slide one last time,
again the price per leash must be $13
to maximize the profit.
I hope you found this helpful.
