joe buys mugs for $2.50 each. If he sells them for $4, he can sell 60 mugs a week. For every 50 cents increase in price, he sells 6 fewer mugs per week. Determine the price which will maximize Joe's profits.
Here is each case, with the values in this order: Price, mugs sold, profit per mug, Total profit
4 -- 60 -- 1.50 -- 60 x 1.5 = 90
4.5 -- 54 -- 2 -- 54 x 2 = 108
5 -- 48 -- 2.5 -- 48 x 2.5 = 120
5.5 -- 42 -- 3 -- 42 x 3 = 126
6 -- 36 -- 3.5 -- 36 x 3.5 = 126
6.5 -- 30 -- 4 -- 30 x 4 = 120
And then it begins decreasing
So, both 5.5 and 6 per mug give a maximum amount, assuming that the price must be a multiple of 50 cents
let the number of 50 cent increases be n
Now:
number sold = 60
price per mug = 4
after increase:
number sold = 60 - 6n
selling price = 400 + 50n
Profit = Revenue - cost
= (60-6n)(400+50n) - 250(60-6n)
= (60-6n)(400+50n - 250)
= (60-6n)(150 + 50n)
= - 300n^2 + 2100n + 9000
if you know Calculus,
d(profit) = -600n + 2100
= 0 for a max of profit
600n = 2100
n = 3.5
was expecting a whole number, but we can interpret this as
3.5 increases of 50 cents or an increase of $1.75
He should sell them at $5.75
If you don't know Calculus, find the vertex of the corresponding
quadratic function and it has to come out to the same answer
To determine the price that will maximize Joe's profits, we need to find the price at which the revenue (sales) minus the cost is highest. Let's break it down step by step:
1. Calculate Joe's total cost:
Joe buys mugs for $2.50 each.
So, his total cost per mug is $2.50.
2. Determine the number of mugs Joe can sell at different prices:
Given that Joe can sell 60 mugs a week when he sells them for $4.
For every 50 cents increase in price, he sells 6 fewer mugs per week.
Let's calculate the number of mugs he can sell at different prices:
60 mugs/week at $4
60 - 6 = 54 mugs/week at $4.50
54 - 6 = 48 mugs/week at $5
48 - 6 = 42 mugs/week at $5.50
...and so on.
3. Calculate Joe's revenue at different prices:
Given the number of mugs he can sell at different prices, we can calculate Joe's revenue by multiplying the price and the quantity:
Revenue at $4 = $4 * 60
Revenue at $4.50 = $4.50 * 54
Revenue at $5 = $5 * 48
Revenue at $5.50 = $5.50 * 42
...and so on.
4. Calculate Joe's profit at different prices:
To calculate Joe's profit, we need to subtract his total cost (Step 1) from his revenue (Step 3) at different prices:
Profit at $4 = Revenue at $4 - (Total cost per mug * Quantity at $4)
Profit at $4.50 = Revenue at $4.50 - (Total cost per mug * Quantity at $4.50)
Profit at $5 = Revenue at $5 - (Total cost per mug * Quantity at $5)
Profit at $5.50 = Revenue at $5.50 - (Total cost per mug * Quantity at $5.50)
...and so on.
5. Identify the price that maximizes Joe's profit:
Compare the profits calculated in Step 4 to determine the price that results in the highest profit for Joe. The price that maximizes his profit will be the one with the highest value among them.
By following these steps, you can determine the price that will maximize Joe's profits.