A company can produce a toy bear at a cost of $5 per bear. When the company sells the bears at a price of $22, they can sell 480 bears each week. It is estimated that for every $0.50 decrease in the price, they can sell 20 more bears.

Question

A company can produce a toy bear at a cost of $5 per bear. When the company sells the bears at a price of $22, they can sell 480 bears each week. It is estimated that for every $0.50 decrease in the price, they can sell 20 more bears.

Find the price that will maximize the profit for the company.

i don;t know how i got 17.12...but it is wrong
]many thanks

Answer 1

So, if you don't know how you got 17.12, were you just guessing? Why not show your work?

if there are x price increases, then revenue is
(22 - x/2)(480+20x)
and the cost is 5(480+2x)
so profit is
(22 - x/2 - 5)(480+2x) = -x^2 - 206x + 8160
Hmmm. This has a maximum at x = -103
Check my math - where did I go wrong?

Answer 2

To find the price that will maximize the profit, we need to determine the relationship between the price, the number of bears sold, and the profit.

Let's start by gathering the given information:

Cost per bear: $5
Selling price per bear: $22
Number of bears sold per week at the given price: 480
Increase in the number of bears sold for every $0.50 decrease in price: 20

To find the profit, we need to subtract the cost per bear from the selling price per bear:
Profit = Selling price - Cost

Let's calculate the profit for the current price of $22:
Profit = $22 - $5 = $17

Now, let's calculate the profit for a decrease in price by $0.50:
Profit = ($22 - $0.50) - $5 = $16.50 - $5 = $11.50

We can observe that for a $0.50 decrease in price, the profit decreases by $5.50 ($17 - $11.50).

Since the number of bears sold increases by 20 for every $0.50 decrease in price, the profit decrease of $5.50 can be allocated as 20 additional bears sold * $0.50 price decrease = $5.50 decrease in profit.

Now, let's calculate the price at which the profit would be zero to find the price that maximizes the profit:

480 bears/20 bears per $0.50 = 24 * $0.50 decrease in price = $12 decrease in price

$22 - $12 = $10

For a price of $10, the profit would be zero. However, this is not the price that maximizes the profit since a lower price would result in negative profit.

To find the price that maximizes the profit, we need to consider the relationship between the price and the profit decrease. We know that for every $0.50 decrease in price, the profit decreases by $5.50.

To maximize the profit, the profit decrease should be equal to or less than $5 for each $0.50 decrease in price. This means that we shouldn't decrease the price by more than $0.50 per 20 additional bears sold.

So, let's start with the current price of $22 and find the profit decrease for a $0.50 decrease in price:

Profit decrease = $5.50

Now, let's calculate the number of $0.50 price decreases that would result in a profit decrease of $5 or less:

Number of $0.50 decreases = $5 / $5.50 = 0.909 ~ 1 (rounded up)

This means that we should decrease the price by only $0.50 to maximize the profit. Therefore, the price that will maximize the profit for the company is:

Price = $22 - $0.50 = $21.50

Thus, a price of $21.50 per bear will maximize the profit for the company.