The profit p(x) of a cosmetics company in thousands of dollars is given by P(x) = -5x^2 + 400x - 2550, where x is the amount spent on advertising, in thousands of dollars. What amount must be spent on advertising to obtain a profit of at least $4 000 000?
-5x^2 + 400x - 2550 > 4000
5x^2 - 400x + 6550 < 0
x^2 - 80x + 1310 < 0
what are the x-intercepts?
x^2 - 80x + 1310 = 0
x^2 - 80x + 1600 = -1310 + 1600
(x-40)^2 = 290
x-40 = ± √290
x = 40±√290
= appr 57.029 or 22.97
They must spend between $22970 and $57029 to make a profit of 4,000,000
test:
let adv be = 22, 000
P(22) = 3830 < 4000 , not enough
let adv be 25,000
P(25) = 4325 > 4000 , that's good
let adv be 57000
P(57) = 4004 > 4000 , that's still good
let adv be 58000
P(58) = 3830 < 4000 , no good
Wolfram also confirms by conclusion
http://www.wolframalpha.com/input/?i=-5x%5E2+%2B+400x+-+2550+%3E+4000
help
Well, to obtain a profit of at least $4,000,000, the company needs to spend an amount on advertising that will make the profit function P(x) exceed that amount.
Let's set up the equation: -5x^2 + 400x - 2550 ≥ 4000
Now, let me grab my clown calculator and do some math... *tapping on imaginary buttons*
*takes a long pause*
After some hilarious calculations, it seems we have a quadratic equation to solve. I'm afraid, however, it's a bit of a frowny-face situation. This quadratic equation has no real solutions.
That means if the value of x is a real number, it is impossible for the company to achieve a profit of $4,000,000.
But hey, who needs a profitable cosmetics company when you can have a good laugh with Clown Bot, right?
To find the amount that must be spent on advertising to obtain a profit of at least $4,000,000, we need to set up and solve an inequality using the given profit function.
The profit function is P(x) = -5x^2 + 400x - 2550.
We want to find the value of x that gives us a profit of at least $4,000,000, so we have the inequality:
P(x) ≥ 4,000,000
Substituting the profit function, we get:
-5x^2 + 400x - 2550 ≥ 4,000,000
Rearranging the inequality, we have:
-5x^2 + 400x - 4,002,550 ≥ 0
To solve this quadratic inequality, we can:
1. Multiply the entire inequality by -1 to make the coefficient of the leading term positive. This changes the direction of the inequality.
5x^2 - 400x + 4,002,550 ≤ 0
2. Factor the quadratic expression if possible.
Since the quadratic expression does not factor smoothly, we can use the quadratic formula to find the x-intercepts.
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac))/(2a)
For this quadratic equation, a = 5, b = -400, and c = 4,002,550.
Plugging in the values into the quadratic formula, we have:
x = (-(-400) ± √((-400)^2 - 4 * 5 * 4,002,550)) / (2 * 5)
Simplifying further:
x = (400 ± √(160,000 + 80,101,000)) / 10
x = (400 ± √80,261,000) / 10
To simplify the equation further, we need to find the value of √80,261,000.
√80,261,000 = √(1000 * 80,261) = (√1000) * (√80,261) = 10 * (√80,261)
Substituting this back into the equation, we have:
x = (400 ± 10 * √80,261) / 10
Simplifying:
x = 40 ± √80,261
So, we have two possible values for x: x = 40 + √80,261 and x = 40 - √80,261.
Therefore, the amount that must be spent on advertising to obtain a profit of at least $4,000,000 is approximately 40 + √80,261 and 40 - √80,261.
To find the amount that must be spent on advertising to obtain a profit of at least $4,000,000, we need to set up an inequality based on the given profit function.
The profit function is given as: P(x) = -5x^2 + 400x - 2550
We want to find the value of x when the profit is at least $4,000,000, so we can set up the following inequality:
P(x) ≥ 4,000,000
Substituting the profit function into the inequality:
-5x^2 + 400x - 2550 ≥ 4,000,000
Now, we need to solve this quadratic inequality. To do that, let's rearrange the inequality to have all terms on one side:
-5x^2 + 400x - 2550 - 4,000,000 ≥ 0
-5x^2 + 400x - 4,002,550 ≥ 0
To solve this quadratic inequality, we can either use factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula:
The quadratic formula is given by: x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = -5, b = 400, and c = -4,002,550
Using the quadratic formula, we have:
x = (-400 ± √(400^2 - 4(-5)(-4,002,550))) / (2(-5))
Simplifying further:
x = (-400 ± √(160,000 - 80,050,800)) / (-10)
x = (-400 ± √(-79,890,800)) / (-10)
Since the discriminant (√(-79,890,800)) is negative, it means there are no real solutions to this inequality. This indicates that spending any amount on advertising will not result in a profit of $4,000,000 or more for this specific profit function.
Therefore, based on the given profit function, it is not possible to obtain a profit of at least $4,000,000 by spending on advertising.