A souvenir shop sells about 200 coffee mugs each month for 6$ each. The shop owner estimates that for each $0.50 increase in the price, he will sell about 10 fewer coffee mugs per month.

a. How much should the owner charge for each mug in order to maximize the monthly income from their sales?

b. What is the maximum monthly income the owner can expect to make from the mugs?

I've been working on this for around 2 or 3 hours now, its one of my Homework problems, I really don't understand it :/ It'd be extremely appreciated if you could show me how to do it. (Work included as I get confused when teachers jump from one place to another)

NOTE: I already have the function and what x equals which is 4, though I don't know where to go from there :/

It would surely help if you showed how you got there. What you provide is not much help.

If x is the number of price increases, then we have

price p = 6.00 + 0.50x
quantity sold q = 200 - 10x

revenue is price * quantity, so

r(x) = (6.00 + 0.50x)(200-10x)
= 1200+40x-5x^2

This is just a parabola, with its vertex at x=4, as you say.

So, just figure f(4) to get the maximum income.

If you don't understand it, how did you come up with the 4?

Well, after spending about another hour to half an hour comparing this problem to my notes and the other similar problem on this website I finally came up with my answer.

a. 8$ for each mug
b. 1280$

I came up with the 4 by looking at my notes and the other problem on the website similar to this and setting up the equation
(6+0.50x)(200-10x) letting x = # of price increases.
Then I foiled that out and got -5x^2+40x+1200
Then I used the equation x=-b/2a and got x=4
After that I made this post as I was unsure what to do with the 4. Then I looked at my notes and took a guess and did 4*0.5 and got 2. I then added this 2 to the six from the original problem and got 8 which lead to my answer 8$ for each mug. So 8$ was my answer for a.
Then I saw I still did not have a response to my question and I went ahead and tried b. Since every 0.50 price increase the store lost 10 mugs that meant 4 price increases because you have to add 0.50 to 6 each increase and you wanted to get 8, so 4 0.50 price increases as I said. Then I multiplied 4*10 to get the amount of mugs lost and subtracted it from 200. the original mug amount. This gets me 160 and I multiply it by 8 to get the maximum monthly income the owner can expect to make from the mugs (question b) which is $1280.

Would this be correct?

You are correct.

4 was indeed the solution to the equation, but you correctly went on to realize that that meant the actual price was $8/mug.

Good work.

Still, just plugging x=4 into the equation would have given the same answer for the income, since that accounted for the actual price and quantity.

Thank you (;

Also thanks for the tip on the plugging in 4 ;D

Sure! I'd be happy to explain how to solve these problems step by step.

a. To find the optimal price for each mug, we need to determine the price that maximizes the monthly income from sales. Let's break it down into smaller steps:

Step 1: Write an equation for monthly income in terms of the price. Let's call the price increase in dollars as 'x'. The monthly income (I) can be calculated by multiplying the number of mugs sold by the price per mug. Since the number of mugs sold decreases by 10 for every $0.50 increase in the price, we can write the equation as:
I = (200 - 10x) * (6 + 0.5x)

Step 2: Simplify the equation. Multiply the two binomials to get a quadratic equation:
I = (1200 + 100x - 60x - 5x^2)

Step 3: Rearrange the equation in standard form (ax^2 + bx + c = 0):
I = -5x^2 + 40x + 1200

Step 4: To find the optimal price, we need to find the value of x that maximizes the monthly income. This can be done by finding the vertex of the quadratic equation.

Step 5: Use the formula x = -b/(2a) to find the x-coordinate of the vertex. In our equation, a = -5 and b = 40:
x = -40 / (2 * (-5))
x = -40 / (-10)
x = 4

Step 6: Substitute the value of x back into the equation to find the price corresponding to the optimal point:
Price per mug = 6 + 0.5 * 4
Price per mug = 8 dollars

Therefore, the shop owner should charge $8 for each mug to maximize the monthly income.

b. Now let's calculate the maximum monthly income the owner can expect to make from the mugs.

Step 1: Substitute the value of x into the monthly income equation derived earlier:
I = (-5 * 4^2) + (40 * 4) + 1200

Step 2: Simplify the equation to find the maximum income:
I = -5 * 16 + 160 + 1200
I = -80 + 160 + 1200
I = 1280

Therefore, the maximum monthly income the owner can expect to make from selling coffee mugs is $1280.

I hope this explanation helps! Let me know if you have any further questions.