Story Problem help (did I get it right?)

[on3winyoureyes]

To help with the upkeep of Washington State parks, the state sells Discover passes at $30 for a yearly pass. In a recent fiscal year, that created 16.8 million dollars in revenue as 560,000 passes were purchased. This however wasn't enough so the state is considering raising the price of the Discover pass. Analysts assume that every $10 increase in price, 100,000 less passes will be purchased.

a) Define the function that models the total revenue as a function of x, the number of $10 price increases.

b) If prices may increase by any whole dollar amount (not just by tens), find the rate that will maximize the revenue. Round to the nearest dollar.

okay for a. I think that the equation is (560,000-100,000x)*(30+10x)
for b. I think it's $28 (I plugged the function into a calculator and found the rate 2.8 had the highest revenue. 10*2.8 is $28 for the ticket price..

 

[wjm11]

To help with the upkeep of Washington State parks, the state sells Discover passes at $30 for a yearly pass. In a recent fiscal year, that created 16.8 million dollars in revenue as 560,000 passes were purchased. This however wasn't enough so the state is considering raising the price of the Discover pass. Analysts assume that every $10 increase in price, 100,000 less passes will be purchased.

a) Define the function that models the total revenue as a function of x, the number of $10 price increases.

b) If prices may increase by any whole dollar amount (not just by tens), find the rate that will maximize the revenue. Round to the nearest dollar.

okay for a. I think that the equation is (560,000-100,000x)*(30+10x)
for b. I think it's $28

Have you tried graphing your revenue equation and seeing if it agrees with your b) solution? Please show your work for b).

 

[stapel]

To help with the upkeep of Washington State parks, the state sells Discover passes at $30 for a yearly pass. In a recent fiscal year, that created 16.8 million dollars in revenue as 560,000 passes were purchased. This however wasn't enough so the state is considering raising the price of the Discover pass. Analysts assume that every $10 increase in price, 100,000 less passes will be purchased.

a) Define the function that models the total revenue as a function of x, the number of $10 price increases.

b) If prices may increase by any whole dollar amount (not just by tens), find the rate that will maximize the revenue. Round to the nearest dollar.

okay for a. I think that the equation is (560,000-100,000x)*(30+10x)

This isn't actually an "equation", as it has no "equals" sign in it. (They were supposed to have taught you about equations and functions back in algebra.) A "function" will be of the form "y = (something)" or "f(x) = (something)". If you've ever seen either of these forms, try using one of them to complete the function which is required for the answer to this part. (Your expression is fine. Just put it equal to a function name.)

for b. I think it's $28

How did you get this value? What revenue value did you obtain when you used x = -0.2? How does this compare with the current revenue?

Please be complete. Thank you!

 

[Ishuda]

To help with the upkeep of Washington State parks, the state sells Discover passes at $30 for a yearly pass. In a recent fiscal year, that created 16.8 million dollars in revenue as 560,000 passes were purchased. This however wasn't enough so the state is considering raising the price of the Discover pass. Analysts assume that every $10 increase in price, 100,000 less passes will be purchased.

a) Define the function that models the total revenue as a function of x, the number of $10 price increases.

b) If prices may increase by any whole dollar amount (not just by tens), find the rate that will maximize the revenue. Round to the nearest dollar.

okay for a. I think that the equation is (560,000-100,000x)*(30+10x)
for b. I think it's $28

a. is correct.
For b. the allowable x values for the equation changes slightly, i.e. the x in the equation is allowed to be in increments of 0.1 ($1 = 0.1 * $10) instead of just integer increments (a $10 increment in ticket cost). Thus for b, we round x to the nearest tenth not the nearest integer.

A nice thing to remember is that given the two zeros of a quadratic a and b, the maximum/minimum for the quadratic is at
x_m = (a + b) / 2.
The zeros of the function in a. are a = 5.6 [= -560,000 / (-100,000)] and b = -3 [= -30 / 10]. So what is the max/min and what does that mean the ticket price is?