A baseball team plays in a stadium that holds sixty-six thousand spectators.
With the ticket price at nine dollars, the average attendance has been twenty-eight thousand.
When the price dropped to six dollars, the average attendance rose to thirty-three thousand.
Find a demand function D of Q, where Q is the quantity or number of spectators,
and we can assume that D of Q is linear.
So, because D of Q is linear, we'll find the equation of D of Q in slope-intercept
form, or the form Y equals MX plus B.
But because our function is D of Q, we would have D of Q equals M times Q plus B.
So, from the given information, we'll find the slope of the line,
and then from there, we can use point-slope form of a line to find D of Q. But because our function
is D of Q, each ordered pair would be Q comma D of Q. Well, if we want Q comma P for price,
the ticket price of nine dollars and the attendance at twenty-eight thousand, the
ordered pair would not be nine comma twenty-eight thousand. It would be twenty-eight thousand
comma nine. And when the price is six dollars, the attendance is thirty-three thousand,
which means the ordered pair would be thirty-three thousand comma six.
It's important that we have these in the correct order, where the first coordinate is the quantity
or number of spectators. The second coordinate would be D of Q, or the price P. Now that we
have these two ordered pairs, we can find the slope, where the slope is equal to the change
of Y divided by the change of X. So, we can call this X sub one, Y sub one, X sub two, Y sub two,
and therefore the slope would be equal to the change of Y, which would be six minus nine,
divided by the change of X, so this would be thirty-three thousand minus twenty-eight
thousand. And therefore the slope of the line is equal to negative three divided by five thousand.
So, now that we have the slope, we can use this on one of these two points
to find the equation in point-slope form and then solve for Y — in our case,
solve for D of Q. So, if we use this slope and this point,
the equation of our function would be, instead of Y, we'd have D of Q minus Y sub one, or minus
nine, equals the slope, which is negative three divided by five thousand times the quantity Q
minus twenty-eight thousand.
Now let's solve this for D of Q on the next slide. Let's begin by adding nine to both sides.
That would give us D of Q equals
negative three divided by five thousand times the quantity Q minus twenty-eight thousand, plus nine.
Now for our homework, we can leave it in this form here, but I'm also going to show how we
can distribute and combine like terms. But again, this would be one option for our function D of Q.
If we did continue, the next step would be to distribute the fraction here.
So, we'd have D of Q equals negative three divided by five thousand times Q. Next, we have this
negative fraction times negative twenty-eight thousand, which we'll use the calculator for.
Negative three divided by five thousand times negative twenty-eight thousand
is positive sixteen point eight. Let's convert this to a fraction though,
since our slope is a fraction, so math, enter, enter.
So, we have positive eighty-four fifths, or plus eighty-four fifths, plus nine over
one. Our common denominator here is five, so we'll multiply nine over one by five over five.
So, we have D of Q equals negative three divided by five thousand times Q, and this would be plus
eighty-four fifths plus forty-five fifths, which would be one hundred twenty-nine fifths.
So again, we can go ahead and leave the demand function in this form here, or if we want, we
can write it in slope-intercept form, which would be this form here. I hope you found this helpful.