The profit p(x) of a cosmetics company , in thousands of dollars . is given by p(x) =-5t^2=400x-2550, where x is the amount spent on advertising in thousands of dollars
a) determine the maximum profit the company can make
b) dtermine the amount spent on advertising that will result in the maximum profit
c) what amount must be spent on advertising to obtain a profit of at least $ 4 000 000
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
To find the maximum profit, we need to determine the vertex of the quadratic equation given by the profit function. The equation is p(x) = -5x^2 + 400x - 2550.
a) To find the maximum profit, we need to find the vertex of the quadratic equation. The x-coordinate of the vertex, denoted by x = -b/2a, will give us the amount spent on advertising that will result in the maximum profit. In this case, a = -5 and b = 400.
Using the formula, x = -b/2a, we have:
x = -(400) / (2 * (-5))
x = 40
So, the amount spent on advertising that will result in the maximum profit is $40,000.
b) To find the maximum profit, we substitute the x-coordinate of the vertex (x = 40) back into the profit function:
p(40) = -5(40)^2 + 400(40) - 2550
p(40) = -5(1600) + 16000 - 2550
p(40) = -8000 + 16000 - 2550
p(40) = 5450
So, the maximum profit the company can make is $5,450,000.
c) To find the amount that must be spent on advertising to obtain a profit of at least $4,000,000, we substitute p(x) = 4000000 into the profit function:
4000000 = -5x^2 + 400x - 2550
This equation is a quadratic equation. We can rearrange it to solve for x:
5x^2 - 400x + 6550 = 0
To solve this equation for x, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the values a = 5, b = -400, and c = 6550 into the quadratic formula, we have:
x = (-(-400) ± √((-400)^2 - 4(5)(6550))) / (2(5))
x = (400 ± √(160000 - 131000)) / 10
x = (400 ± √29000) / 10
x = (400 ± 170) / 10
So, the two possible solutions for x are:
x₁ = (400 + 170) / 10 = 57
x₂ = (400 - 170) / 10 = 23
Therefore, to obtain a profit of at least $4,000,000, the company must spend at least $57,000 or $23,000 on advertising.
To determine the maximum profit the company can make, we need to find the vertex of the quadratic function p(x) = -5x^2 + 400x - 2550. The vertex represents the maximum point on the graph and gives us the maximum profit.
a) To find the vertex, we can use the formula x = -b / 2a. In this case, a = -5 and b = 400.
First, we need to calculate -b/2a:
-b / 2a = -400 / (2 * -5) = -400 / -10 = 40
So the x-coordinate of the vertex is 40. Now let's find the corresponding y-coordinate (the maximum profit) by substituting x = 40 into the equation p(x):
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 (thousands of dollars).
b) To determine the amount spent on advertising that will result in the maximum profit, we already found the x-coordinate of the vertex in part a. The company should spend $40,000 (thousands of dollars) on advertising to achieve the maximum profit.
c) To obtain a profit of at least $4,000,000 (thousands of dollars), we need to solve the equation p(x) >= 4,000. Let's set up this inequality:
-5x^2 + 400x - 2550 >= 4,000
To solve this quadratic inequality, we need to find the x-values for which the equation is true. This can be done by first rearranging the inequality to have zero on one side:
-5x^2 + 400x - 2550 - 4,000 >= 0
-5x^2 + 400x - 6550 >= 0
Now, we can either factorize or use the quadratic formula. However, let's solve it by factoring. We need to find the values of x that make the quadratic expression non-negative.
-5x^2 + 400x - 6550 = 0
Finding the roots of this equation will give us the critical points where the profit is either equal to or below $4,000,000. We can factor out a common factor, -5:
-5(x^2 - 80x + 1310) = 0
Now, let's find the roots of the quadratic equation within the parentheses:
x^2 - 80x + 1310 = 0
Utilizing factoring, we need to find two numbers that multiply to give 1310 and add up to -80. After some exploration, we find that the numbers are -10 and -70:
(x - 10)(x - 70) = 0
Now, we have two values of x:
x - 10 = 0 --> x = 10
x - 70 = 0 --> x = 70
The profit will be at least $4,000,000 when the amount spent on advertising is either $10,000 or $70,000 (thousands of dollars).