a) You need to sell 12 or 29 dresses.
b) Any number between 2 and 39 dresses will give a positive profit.
How many dresses must be sold in order to make a profit of 3000 euros?
Here we know that the profit, in euros, as a function of the number of dresses sold is:
P(x) = -11*x^2 + 450*x - 800
Now, if you want to find how many dresses you need to sell to have a profit of 3000 euros, then you need to solve:
P(x) = 3000 = -11*x^2 + 450*x - 800
So we need to solve the quadratic equation:
-11x^2 + 450x - 800 - 3000 = 0
-11x^2 + 450x - 3800 = 0
The solutions are given by Bhaskara's formula:
x = (-450 ± √(450^2 - 4*(-3800)*(-11))/2*(-11)
x = (-450 ± 187.9)/(-22)
We have two solutions that give the same profit:
- x = (-450 + 187.9)/-22 = 11.9 that can be rounded to 12.
- x = (-450 - 187.9)/-22 = 28.99 that can be rounded 29.
Then if you sell either 12 or 29 dresses, you will get a profit of 3000 euros.
b) To make a profit you need to sell more than P = 0, so let's solve that first:
P= 0 = -11*x^2 + 450*x - 800
The solutions are:
x = (-450 ± √(450^2 - 4*(-800)*(-11))/2*(-11)
x = (-450 ± 409)/(-22)
The smaller solution is:
x = (-450 + 409)/-22 = 1.86 that can be rounded to 2.
(because you can't sell 1.86 of a dress)
The other solution is:
x = (-450 - 409)/-22 = 39
So, between 2 and 39 dresses, you will make a profit.
If you want to learn more about quadratic equations:
https://brainly.com/question/1214333
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