In this question,
we're asked to find the practical domain
and practical range based upon the given situation
where it says you purchased 90 ice cream cones
for $18 and plan to sell them at the park
for 82 cents each, your profit can be determined
by the function p of x equals zero point eight two x
minus 18, where p represents your profit
and x represents number of ice cream cones sold.
So the first thing we should recognize
is that the profit function is a linear function
in slope intercept form, meaning it fits the form
f of x equals mx plus b, so notice how
the y intercept or vertical intercept
is negative 18 and the slope is zero point eight two.
So if you were to graph the linear function
on the coordinate plane, it would look like this.
If we ignored the context of the situation,
because the graph moves left and right
without any holes or breaks, the domain
would be all real numbers, and because
the graph moves down and up indefinitely
without any holes or break, the range would also be
all real numbers, but again, because of the context
of the situation, this would not be
the practical domain and practical range.
Let's go back and look at our problem.
Remember x or the input represents
the number of ice cream cones sold
and you purchased 90 ice cream cones for $18,
so the least number of cones you could sell
would be zero, you can't sell a negative number of cones,
and the most number of cones you could sell
would be 90 because that's how many you bought,
which means the practical domain
would be when x is greater than or equal to zero
and less that or equal to 90.
For review, if they wanted to express
the domain using interval notation,
we'd use a square bracket zero comma 90 square bracket.
Remember, the square brackets indicate
the end points are included.
Now let's go back to our graph for a moment.
Now that we know the practical domain,
let's go ahead and indicate this on the x axis.
The practical domain is the closed interval
from zero to 90, so of course,
if we restrict the domain, it's going to affect the range.
Notice how the function value of profit,
when x is zero, would be the y intercept here.
This would be the lower bound of the practical range.
The upper bound of the range would be the profit
when x equals 90, which would be approximately here.
So this would be the practical range.
Again, the lower bound is the y intercept,
which we know is negative 18.
To find the upper bound of the range,
we'd have to substitute x equals 90
into our profit function.
Let's go ahead and show how to find both of these.
To find the lower bound of the practical range,
we have to find the function value p of zero.
Again, p of zero is going to give us the y intercept.
So this tells us that if you sell zero ice cream cones,
you're going to lose $18, because remember,
that's how much you paid to purchase the 90 ice cream cones.
And to find the upper bound, we need to find the profit
when you sell all 90 ice cream cones,
so p of 90 would be equal to zero point eight two
times 90 minus 18, and I've already determined this
to save some time, it's 55.8, which does represent $55.80,
but this would be the upper bound of our practical range.
So again, this does tell us that if we sell zero
ice cream cones, we're actually going
to have a loss of $18, if we sell
all 90 ice cream cones, we'd have
our maximum profit of $55.80.
I hope you found this helpful.