Gilberto S. answered • 09/02/20
Experienced College Professor
The Profit = Revenue - Cost, so the Profit = (20x) - (5600 + 10x + 0.002x^2)
We can distribute this
Profit = 20x - 5600 - 10x - 0.002x^2 and we can combine like terms and rearrange the terms by degree..
Profit = -0.002x^2 + 10x - 5600
Now we want to find which production level guarantees a profit of 1120. So we want to then solve:
Profit ≥ 1120
-0.002x^2 + 10x - 5600 ≥ 1120 we will subtract 1120 from both sides;
-0.002x^2 + 10x - 6720 ≥ 0 then to make things easier for ourselves, we will multiply both sides by 1000 to remove the decimal point:
-2x^2+ 10000x - 6720000≥ 0. then we can divide both sides by -2 (keep in mind that if we multiply or divide an inequality by a negative number , we switch the direction of the inequality)
x^2 -5000x +3360000 ≤ 0
To solve this inequality we need to make sure that we have a 0 on one side (which we have in this case). Then we will figure out when the left hand side =0. And we know that for each interval in between those zeroes, the left hand expression will either be positive or negative.
Let's solve x^2 -5,000x +3,360,000 =0 we can try to factor the left hand side but in this case it will be easier to use the quadratic formula. where a = 1, b = -5000 and c = 3360000.
When we plug in those values we get x = (5000 ± √(11560000))/2 = (5000 ± 3400) /2 = 1600/2 or 8400/2 = 800 or 4200
These two points divide the real line in to three intervals: The values less than 800, the values between 800 and 4200 and the values greater than 4200. We can use test points or a graphing program to check that in the middle interval the profit is at least 1120. Also AT the end points, the profit equals 1120.
So the final answer should be [800, 4200]
