Question

Hi! Please need help with this pre-calc problem!

Answer 1

Each unit of brand awareness of a product generated by an ad campaign costs $.10 more than the previous one (I.e., the first costs .1, the second costs .1 more and is .2, the third costs .1 more, and is .3, etc.), and the overhead cost of $50 is needed to begin the ad campaign. Each unit of awareness generates exactly $5 in revenue.

Find a function that models this in terms of awareness and profit. Graph it.

How much awareness gets us to break even?

How much maximizes the profit?

Answer 2

The only real obstacle in this problem, other than understanding what it is asking, is the non-continuous nature of the solution. In other words, we CANNOT get 10.5 units of awareness. It must be 10 or 11. We cannot get 5.38 units of awareness. It must be 5 or 6 units.

Do you see that this must be so?

Answer 3

I do see this. Just dont know how to approach it

Answer 4

Got it. I do understand that. That part is actually simple when weitten like that

Answer 5

Perfect. It's important to observe that up front so it doesn't confuse us later.

We'll need to be able to add up an Arithmetic Series. That's the way Awareness is sold, apparently.

In particular, we need to find a more convenient expression for this sum.
0.1 + (0.1+0.1) + (0.1+2*0.1) + (0.1+3*0.1) + ... + (0.1+(n-1)*0.1)

Stare at that a little and make sure you know where it came from.

Answer 6

Now, a little algebra

0.1 + (0.1+0.1) + (0.1+2*0.1) + (0.1+3*0.1) + ... + (0.1+(n-1)*0.1)

0.1*n + 0.1(0 + 1 + 2 + ... + (n-1))

Do you still believe?

Answer 7

Got it

Answer 8

Or, since the starting value and common difference are the same, we could have written

0.1(1 + 2 + 3 + 4 + ... + n)

That's actually quite a lot more convenient.

Do you know the sum of the first n consecutive Natural Numbers? It's an important one to remember.

Answer 9

Is the first number just .1 based on the above equation?

Answer 10

Yes. All that is left is to add up the Natural Numbers.

Answer 11

Ok. Just getting stuck on the "+n"

Answer 12

Each term contains 0.01 from the initial value. That makes 0.01*n

Anyway, 1 + 2 + 3 + ... + n = n*(n+1)/2

Try it out: 1+2+3+4 = 10 4*5/2 = 20/2 = 10

Your task is now to put all the pieces together.

\(0.01\cdot\dfrac{n(n+1)}{2}\) is the total cost of generating awareness

5n is the total revenue from the venture.

50 is the fixed cost.

\(5n - 0.01\cdot\dfrac{n(n+1)}{2} - 50 = 0\) is the break even

\(5n - 0.01\cdot\dfrac{n(n+1)}{2} - 50 > 0\) is the profitable