Calculus Trouble

[V|P]

We're doing optimization in AP Calc (AB) and I'm completely lost on one type of problem. Here's two examples I have:


An airline finds that if it charges $400 to fly to london, it will sell 500 tickets per day. It estimates that each $10 price reduction will result in 25 more tickets sold per day. Find the ticket price that will maximize the airline's revenue. How many tickets will be sold at this price?



Suppose that the relationship between the tax rate, t, on imported handbags and the total sales, S, is given by S(t) = 100 - 160^(1/4), where S is measured in millions of dollars. Find the tax rate t that maximizes revenue for the government.


The answers would definitely help, but a step by step walk through for one or both of the problems would be much more useful to me since I have many more to do. Any help is appreciated. Thanks!

[shwaip]

your second problem is missing something.

[_k_]

Problem 1:
An airline finds that if it charges $400 to fly to london, it will sell 500 tickets per day.
500 base tickets per day
400 base price of a ticket

It estimates that each $10 price reduction will result in 25 more tickets sold per day. Find the ticket price that will maximize the airline's revenue.
-10 price change of the base ticket price
+25 change in ticket sale from base value

Technically you have to equations here.
Base Ticket Sales + (change in ticket sales*x) 500 + 25x =
Base Ticket Price + (change in ticket price*x) 400 - 10x =

They both yield a line that can be graphed. You can do this on a graphing calculator and get the intersection that way but the paper and pencil method is what you need to do.
We find the intersection of these two equations by setting them equal to each other and solving for the single variable.
500 + 25x = 400 - 10x
Take the absolute value of your answer.
-----------------------------------------------------------------------------
The second problem it seems you forgot to give one of the equations, the equation for t. I think I am right for how I explained the question.

[V|P]

The second problem it seems you forgot to give one of the equations, the equation for t. I think I am right for how I explained the question.

After staring at the problem for an hour, that's what I thought, but we've all been forgetting what the tax rate is. The given equation is for total sales, while the revenue (R) for the GOVERNMENT is the tax (t) collected FROM the total sales. So, they give you one equation and the second must be R = S*t or Revenue is given by sales multiplied by the tax rate. After solving through, I got .0625 for t, or 6.25% which seems like a reasonable tax rate so I'm assuming it's correct.

Thanks for the walkthrough for the other problem, but I actually finished my entire assignment already. I did it slightly differently, using three variables and three equations:
# of tickets sold: y
$ of ticket: x
# of price reductions: z

R = xy
y=500+25z
x=400-10z

[Nomad]

Just start with adding a variable:

x = price reduction
n = number of tickets sold
p = price of the ticket (which is 400 - x)


The increase of tickets sold is 25 for every $10, or 25/10.

Then the number sold n = (500 + 25x) and the price per ticket is p = (400 - 10x)

The maximization of revenue is found at R(x) = (500 + 25x)*(400 - 10x)

Take the derivative (absolute value)

*edit*

someone posted when I did this

[shwaip]

you need to make sure the second derivative is positive.