The demand equation for a certain product
is given by p equals 138 minus 0.065x,
where p is the price in
dollars of the product
and x is the number of units produced.
The total revenue obtained by producing
and selling x units is
given by R equals x times p.
We're asked to determine
the prices that would yield
a revenue of $7,770.
So for our first step let's
write the revenue function
as a function of x.
So we'll substitute 138 minus 0.065x
for p in the revenue equation.
So that would give us
revenue equals x times
138
minus
0.065x.
And now our goal here is
to determine the price
when the revenue is $7,770.
So now we'll make a substitution for R.
We'll substitute 7,770 for R.
And now we have an
equation with one variable.
We have an equation with only x.
So we'll solve this equation
for x, but just keep in mind
our goal here is to find the price
that would yield a revenue of $7,770.
So once we have solved this equation for x
we'll have to come back and find p
using this equation here.
So for our step in our
equation was to distribute x.
So we have 7,770 equals,
let's write the x-squared term first.
So negative 0.065x squared
and then plus
138x.
Because we have a quadratic equation
let's set it equal to zero.
So we'll subtract 7,770
from both sides of the equation.
So now we have zero equals
negative 0.065x squared
plus 138 x
minus 7,770.
We're not going to be able to factor this.
So to solve this quadratic
equation we'll be using
the quadratic formula
shown here for review,
where a is the coefficient
of the x-squared term
so a equals negative 0.065.
B is equal to coefficient
of x which is 138.
And c is the constant term
which is negative 7,770.
And now we'll substitute a, b, and c
into the quadratic formula.
Let's do this on the next slide.
We'll have x equals.
In the numerator we have negative b,
that would be negative 138 plus or minus
the square root
of b squared, that'd be 138 squared
minus four times a which is negative 0.065
times c which is negative 7,770.
All this is divided by two times a
which is two times negative 0.065.
Now let's go to the calculator
and determine our two solutions.
We'll have two solutions because
of the plus or minus here.
Let's let the first solution be x sub one
and the second solution be x sub two.
We want the entire numerator
in a set of parentheses.
So we'll have open
parenthesis for the numerator
and then negative 138.
Let's first use the plus
sign, so plus square root.
And then we have square root of 138
squared
minus four
times
negative .065
times
negative 7,770.
Right arrow to get outside
of the square root.
Closed parenthesis for the numerator.
And then divide it by, we
also want the denominator
in a set of parentheses.
So open parenthesis two times
negative .065,
closed parenthesis and closed parenthesis.
And Enter.
So to four decimal places x sub one
is approximately 57.8824.
Now to find x of two we just
need to make one change.
We need to change this addition sign
to a subtraction sign.
So instead of entering all this again
we can press Second + Enter
which brings up the previous expression.
And now we can arrow back and
edit anything that we want.
We only have one change here.
We want to change, again, the plus sign
to a subtraction sign here.
And Enter.
So x sub two is 2065.1945.
Now take these two solutions
back to the previous slide
so you can actually find the price.
Remember, x is not the price,
x is the number of units produced.
So now we use the price
equation to answer the question
about what the lowest price
and highest price would be
to yield revenue of $7,770.
So first we use x sub
one so we'd have p equals
138
minus 0.065
times
57.8824.
So we'll call this p sub one.
And p sub two would be equal to 138
minus 0.065
times 2065.1945.
And now we'll go back to the calculator.
Because this is money we will
round to the nearest cent.
So for p sub one we'll have 138
minus .065
times 57.8824.
So for the nearest cent
we would have $134.24.
And now we'll find p sub two.
So again, if we press Second + Enter
we can just edit the
entry here in parentheses
which for p sub two is 2065.1945.
So p sub two is $3.76.
So if the product is priced
at either of these two amounts
the revenue will be $7,770.
And we're asked to enter
the lowest price here
and the highest price here.
The units of dollars are already provided,
so we'll enter 3.76 here
and 134.24
here.
I hope you found this helpful.
