A sporting goods store sells 90 ski jackets in a season for $200 each. Each $10 decreases in the price would result in 5 more jackets being sold. Find the number of jackets sold and the selling price to give revenues of $17,600 from sales of ski jackets
Let's start by assuming that the number of jackets sold at $200 each is denoted by n. The total revenue from these sales would then be 200n.
The number of jackets sold can be increased by decreasing the price by $10, resulting in an increase in the number of jackets sold by 5. This means that for each decrease of $10 in the price, the number of jackets sold increases by a factor of 5.
Let's say the price is decreased by $10 m times. This would result in an increase in the number of jackets sold to n + 5m.
The new selling price after decreasing the price by $10 m times would be $200 - $10m.
The new revenue from these sales would be (200 - 10m)(n + 5m).
Given that the total revenue is $17,600, we can set up the following equation:
(200 - 10m)(n + 5m) = 17,600
Expanding the equation, we get:
200n + 1000m - 10mn - 50m^2 = 17,600
Rearranging the terms, we get:
-10mn - 50m^2 + 200n + 1000m = 17,600
Now, we can solve for n and m.
Since we have two variables, we need another equation to solve the system. We can use the fact that initially, 90 jackets were sold at $200 each.
This gives us the equation:
200n = 90(200)
Simplifying, we get:
200n = 18,000
Dividing by 200, we get:
n = 90
Now we can substitute this value of n back into the first equation:
-10mn - 50m^2 + 200n + 1000m = 17,600
-10(90)m - 50m^2 + 200(90) + 1000m = 17,600
-900m - 50m^2 + 18,000 + 1000m = 17,600
-50m^2 + 100m + 18,000 = 17,600
-50m^2 + 100m + 400 = 0
Dividing the equation by -50, we get:
m^2 - 2m - 8 = 0
Factoring the equation, we get:
(m - 4)(m + 2) = 0
Setting each factor equal to 0, we find two possible values for m:
m - 4 = 0 --> m = 4
m + 2 = 0 --> m = -2
Since the number of jackets sold cannot be negative, we discard the solution m = -2.
Therefore, the number of jackets sold, n, is 90 and the number of $10 decreases in the price, m, is 4.
To find the selling price, we substitute the value of m into $200 - $10m:
Selling price = $200 - $10(4) = $200 - $40 = $160
Thus, to give revenues of $17,600, the sporting goods store needs to sell 90 jackets at a price of $160 each.