The profit p(x) of a perfume 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
The vertex of a parabola y=ax2 + bx + c occurs at x = -b/2a
So, when x = 40, we have p(x) = 5450
Duh. We just said that x = 40. How many dollars does that represent?
Now we want -5x2+400x-2550 >= 4000
Think of the parabola. It opens downward. So, the portion above the line y=4000 is just a small cap, where
22.97 <= x <= 57.03
To answer these questions, we need to understand the concept of finding the maximum value of a quadratic equation. The equation for the profit is given as p(x) = -5x^2 + 400x - 2550.
a) To determine the maximum profit the company can make, we need to find the vertex of the parabolic function. The vertex gives us the highest point on the curve, which represents the maximum value. The equation for the x-coordinate of the vertex is given by x = -b/2a, where a, b, and c are the coefficients of the quadratic equation.
In this case, a = -5, b = 400, and c = -2550. Let's substitute these values into the formula:
x = -400 / (2 * (-5))
x = -400 / (-10)
x = 40
So, the x-coordinate of the vertex is 40. To find the maximum profit, we substitute this value of x back into the profit equation:
p(x) = -5(40)^2 + 400(40) - 2550
Evaluating this expression, we get:
p(x) = -5(1600) + 16000 - 2550
p(x) = -8000 + 16000 - 2550
p(x) = 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 found the x-coordinate of the vertex to be 40. This means that when the company spends $40,000 on advertising, it will 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 equal to 4,000,000 and solve for x. Let's set up the equation:
-5x^2 + 400x - 2550 = 4,000,000
Rearranging the equation to standard form:
5x^2 - 400x + 4250 = 0
Now, we can use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = 5, b = -400, and c = 4250. Plugging these values into the formula, we get:
x = (-(-400) ± √((-400)^2 - 4 * 5 * 4250)) / (2 * 5)
x = (400 ± √(160000 - 85000)) / 10
x = (400 ± √75000) / 10
Simplifying further:
x = (400 ± 274.57) / 10
So, x can have two values:
x = (400 + 274.57) / 10 = 67.457
x = (400 - 274.57) / 10 = 12.543
Therefore, the amount that must be spent on advertising to obtain a profit of at least $4,000,000 is between $12,543 and $67,457 (in thousands of dollars).