The Difficult Math Problem

They said the quest was going to be easy. That getting the All Knowing Calculator would be a breeze, but they were wrong. It was easy to get past all the booby traps and the lava river, but now is where we really get tested. We stood in front of a huge door with lava pouring to the sides of it. In the middle of the door was a screen. It said:

To get past this door, you must solve a difficult math problem. Do you accept?

On the bottom of the screen were two buttons. One was green and said accept, and the other was red and said decline. Without thought, I pressed accept. The screen went blank and then a sentence fazed on the screen. It said:

Choose one person to solve the problem. Once you have chosen, have the person press the red button in the middle of the screen.

Everyone looked at me. I did not want to do it. What if it was really hard? Although, by the looks of my group I would not be able to get out of this one. I pressed the button, and out of nowhere walls surrounded me and a roof appeared above me. A seat sprouted from the ground forcing me to sit on it. Then a desk landed in front of me with a pencil, a piece of paper, and a calculator on it. The screen was still in the room and the math problem had appeared. It said:

~An office supply store sells about 80 graphing calculators per month for $120 each. For each $6 decrease in price, the store expects to sell eight more calculators. The revenue from calculator sales is given by the function R(n) = (unit price)(units sold), or R(n) = (120 - 6n)(80 + 8n), where n is the number of $6 price decreases.

a.How much should the store charge to maximize monthly revenue?

b.Using a different revenue model, the store expects to sell five more calculators for each $4 decrease in price. Which revenue model results in a greater maximum monthly revenue? Explain.~

Once you have solved the problem, type it into the answer box.

Oh no, I thought to myself. This will take awhile.

As if the screen read my mind, a timer appeared on the top right screen. It said 10 minutes to solve. It started to count down.

That's not fair, I thought.

I reread the question and I realized what I had to do. I started by rewriting the function: R(n)= (120 - 6n)(80 + 8n). To make it easier, I knew I would have to write in standard form. I got there by using the foil method which looked like this: (120)(80)+(120)(8n)+(-6n)(80)+(-6n)(8n). Once I finished my calculations, I got: R(n)=-48n2+480n+9600. There was eight and a half minutes left.

I now had to find the vertex. I tried to remember how to find the vertex. All I needed for part A was the x-value of the vertex. It took me a few seconds to remember that it was -b over 2a. I plugged 480 into B and-48 into A. I got -480 over -96 which equals 5. I realized that I got the first part of the problem and I still had 7 minutes left. I typed in the answer box that said part a on the screen:

~The store should charge 5 $6 price decreases.~

After I typed the answer in, the part a answer box turned green and said correct. I was relieved. I did not want to know what happens if I get it wrong.

I reread the part b question and I figured out that the question it was asking me was to change the original function. All I had to do was change the numbers in front of n. The new function was: R(n)= (120-4n)(80+5n). To make it easier I wrote it in standard form which I got by using the foil method: (120)(80)+(120)(5n)+(-4n)(80)+(-6n)(5n). Which equals: -20n2+280n+9600. There was 5 minutes left.

I have time, I thought.

Yet again, as if the computer read my mind. Red lights started to flash in the room. This was really distracting. I had to solve this. I started solving the x-value of the vertex. I plugged 280 into B and -20 into A. Which looked like: -280 over -40 which equals 7. The question was asking which revenue would have a greater monthly revenue, so I will have to plug in each x-value into their function. There was 4 minutes left.

The chair started to vibrate. Making it harder to write. I tried to write on the paper, but it just came out all squiggly. I had to improvise with it. I put 5 into n on the first function so it looked like: R(n)=-48(5)2+480(5)+9600. Then I plugged 7 into the second function so it looked like: R(n)=-20(7)2+280(7)+9600. From that I had: -48(25)+2400+9600 for the first function, and: -20(49)+1960+9600. Which turned out to be: R(n)=10800 for the first function and: R(n)=10580 for the second function. I got my answer and there was 2 minutes left.

I tried to touch the screen on the wall, but as I reached for it, the wall backed up. I got off the chair and walked towards the screen. When I tried to reach for the screen, it backed up again.

This isn't good, I thought.

I ran towards the screen and it kept going backward. I had to get to it. When I started to get close to it, it moved faster. After about 30 seconds of running, I stopped to catch my breath. The wall stopped after it had gone a pretty good distance away from me and I made a run for it. Wind began to push me as I ran. With all my might, I finally got to the screen. I had 30 seconds left. I needed to write an answer that would make sense.

~I wrote in the part b answer box:

The model revenue of a $6 decrease in price has a greater maximum monthly revenue than a $4 decrease. The $6 decrease had a $10800 maximum value. While the $4 decrease had a $10580 maximum monthly revenue.~

There was 5 seconds left. I had to hope this answer was correct. I pressed the button and it said thinking on the screen. After a few seconds it said correct and I was really relieved.

The walls around me lifted and the screen disappeared. I looked at my group and they looked really pleased. They patted me on the back as the door opened, and we walked towards the All Knowing Calculator. The thing that would give us the knowledge of all math problems.

THE END