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

a) determine the max profit the company can make
b)Determine the amount spent on advertising that will result in the max profit
c)What amount must be spent on advertising to obtain a profit of at least $4 000 000

p(x) = -5x^2 + 400x - 2550

p'(x) = -10x + 400
= 0 for a max of p(x)
10x = 400
x = 40 -----> answer for b)

p(40) = -5(40^2) + 400(40) - 2550 = 5450 -- > a)

for c)
solve
4000 = -5x^2 + 400x - 2550

The profit of a cosmetics company, P= -5x^2+40x-20

a) The maximum profit can be determined by finding the vertex of the quadratic equation. Since the coefficient of the x^2 term is negative, the vertex represents the maximum point. The x-coordinate of the vertex can be found using the formula x = -b/(2a), where a, b, and c are the coefficients in the quadratic equation.

In this case, a = -5, b = 400, and c = -2550. Using the formula, we have:
x = -400/(2*(-5))
x = -400/(-10)
x = 40

To find the maximum profit, substitute the value of x into the quadratic equation:
p(40) = -5(40)^2 + 400(40) - 2550
p(40) = -5(1600) + 16000 - 2550
p(40) = -8000 + 16000 - 2550
p(40) = 5450

Therefore, the maximum profit the company can make is $5,450,000.

b) To determine the amount spent on advertising that will result in the maximum profit, we already calculated that x = 40. Therefore, the company should spend $40,000 on advertising to achieve the maximum profit.

c) To find the amount that must be spent on advertising to obtain a profit of at least $4,000,000, we need to set the profit equation greater than or equal to $4,000,000 and solve for x.

-5x^2 + 400x - 2550 ≥ 4000

Since it is a quadratic inequality, we will solve it by first rewriting it:

-5x^2 + 400x - 6550 ≥ 0

To find the values of x, we can factor the quadratic equation:

(-5x - 530)(x - 20) ≥ 0

From this, we can see that the inequality is satisfied when:
-5x - 530 ≤ 0 and x - 20 ≥ 0

Solving these inequalities separately:
-5x ≤ 530 and x ≥ 20

Dividing both sides of the first inequality by -5, we have:
x ≥ -106

Combining this with x ≥ 20, we get:
x ≥ 20

Therefore, the amount that must be spent on advertising to obtain a profit of at least $4,000,000 is $20,000.

To find the maximum profit, we need to maximize the profit function p(x) = -5x^2 + 400x - 2550.

a) To determine the maximum profit the company can make, we can use the concept of vertex of a parabola. The vertex of a parabola represents the maximum or minimum point. In this case, since the coefficient of x^2 is negative, the graph of the parabola opens downwards, indicating a maximum point.

The x-coordinate of the vertex can be calculated using the formula x = -b/2a, where a and b are the coefficients of x^2 and x, respectively. In our case, a = -5 and b = 400.

x = -400 / (2 * -5) = -400 / -10 = 40

Now, substitute the x-coordinate of the vertex into the profit function to find the maximum profit:

p(40) = -5(40)^2 + 400(40) - 2550
= -5(1600) + 16000 - 2550
= -8000 + 16000 - 2550
= 5450

Therefore, the maximum profit the company can make is $5,450,000.

b) To determine the amount spent on advertising that will result in the maximum profit, we use the x-coordinate of the vertex, which is x = 40. So, the company should spend $40,000 on advertising to achieve the maximum profit.

c) To find the amount that must be spent on advertising to obtain a profit of at least $4,000,000, we need to solve the equation p(x) ≥ 4,000,000.

-5x^2 + 400x - 2550 ≥ 4,000,000

To solve this quadratic inequality, we need to rearrange it and set it to zero:

-5x^2 + 400x - 2550 - 4,000,000 ≥ 0
-5x^2 + 400x - 4,002,550 ≥ 0

Now, we can factor this quadratic equation or use the quadratic formula to find the roots. However, since we are only interested in the range of values for x that satisfies the inequality, it's easier to use a graphing calculator or software to find the x-intercepts of the quadratic equation. Setting the equation equal to zero (using an inequality sign) gives us the values of x that satisfy the inequality.

Using a graphing calculator or software, we find that the x-intercepts are approximately x ≈ 368.552 and x ≈ 71.448.

Therefore, the amount that must be spent on advertising to obtain a profit of at least $4,000,000 is between $71,000 and $368,000.

a. Complete the square.

-5x^2 + 400x - 2550
= -5(x^2 - 80x) - 2550
= -5(x^2 - 80x + 1600 - 1600) - 2550
= -5(x + 40)^2 + 8000 - 2550
= -5(x + 40)^2 + 5450
5450 x 1000 = $5,450,000
b. x = 40
40 x 1000 = $40000
c. Between $22 971 & 57 029