a souvenir shop sells about 300 coffee mugs per month for $7.50 each. the shop owner estimates that for each $0.75 increase in the price, he will sell about 15 fewer coffee mugs per month.
a. how much should the owner change 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?
if there are x 75-cent increases, then there are 300-15x mugs sold.
So, now we have the revenue as
r(x) = (7.50 + 0.75x)(300-15x)
= -11.25x^2 + 112.5 + 2250
Now just find the vertex of the parabola.
I've never understood how to do these problems, i wonder if i could get help with this one and i'll use it as an example to solve the similar problems.
Just as a reminder, revenue is price * demand. That is, toe total income for all the items sold.
recall that for y = ax^2+bx+c the vertex is at x = -b/2a. So, for this one, the vertex is at
x = 112.5/22.5 = 5
Since the parabola opens downward, the vertex is a maximum. So, the maximum revenue is
r(5) = 2531.25
Don't forget all your algebra I, now that you're taking algebra II.
To find the optimal price per mug and the maximum monthly income, we need to understand the relationship between the price and the number of mugs sold, as well as the relationship between the number of mugs sold and the monthly income.
Step 1: Determine the relationship between the price and the number of mugs sold.
The problem states that for each $0.75 increase in the price, the shop owner sells about 15 fewer mugs per month. This implies that there is a linear relationship between the price and the number of mugs sold. We can use the formula:
Number of mugs sold = Initial number of mugs sold - (Number of $0.75 increases × 15)
Let's calculate the initial number of mugs sold:
Initial number of mugs sold = 300 (given)
Step 2: Calculate the number of mugs sold at different price points.
Let's assume the initial price is $7.50 per mug.
For $7.50 per mug:
Number of mugs sold = 300 - (0 × 15) = 300
For $7.50 + $0.75 = $8.25 per mug:
Number of mugs sold = 300 - (1 × 15) = 285
Similarly, we can calculate the number of mugs sold for different price points.
Step 3: Calculate the monthly income for different price points.
The monthly income is calculated by multiplying the price per mug by the number of mugs sold.
For $7.50 per mug:
Monthly income = $7.50 × 300 = $2,250
For $8.25 per mug:
Monthly income = $8.25 × 285 = $2,353.75
We can repeat this process and calculate the monthly income for different price points.
Step 4: Determine the optimal price and maximum monthly income.
To maximize the monthly income, we need to find the price point that generates the highest income.
Let's tabulate the prices and the corresponding monthly incomes:
Price per mug | Monthly income
----------------------------------
$7.50 | $2,250
$8.25 | $2,353.75
... | ...
By comparing the monthly incomes, we can identify the price point that generates the highest income. This will be the optimal price per mug.
Once we determine the optimal price, we can calculate the corresponding number of mugs sold and the maximum monthly income.
You can follow these steps to solve part (a) and part (b) of the question.