The profit P(x) of a cosmetics company, in thousands of dollars, is given by 𝑃(𝑥) = −5𝑥2 + 400𝑥 +2500, 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?

Question

The profit P(x) of a cosmetics company, in thousands of dollars, is given by 𝑃(𝑥) = −5𝑥2 + 400𝑥 +2500, 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?

Answer 1

clearly, you just have to solve

−5x^2 + 400x +2500 ≥ 4000
so crank 'er out. Post your work if you get stuck.

Answer 2

I need help

Answer 3

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 using the profit function.

The profit function is given as P(x) = -5x^2 + 400x + 2500

We want to find the value of x when the profit P(x) is greater than or equal to $4,000,000. This can be written as:

P(x) >= 4,000,000

Substituting the given profit function, we have:

-5x^2 + 400x + 2500 >= 4,000,000

Rearranging the inequality, we have:

-5x^2 + 400x + 2500 - 4,000,000 >= 0

Next, we simplify the inequality:

-5x^2 + 400x - 3,997,500 >= 0

Now, we can solve this inequality to find the range of values for x.

Step 1: Find the x-intercepts by setting the equation equal to zero:

-5x^2 + 400x - 3,997,500 = 0

Step 2: Use factoring, completing the square, or the quadratic formula to solve for x. In this case, we'll use the quadratic formula:

The quadratic formula is given by:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)

For our equation, with a = -5, b = 400, and c = -3,997,500, we have:
x = (-400 ± sqrt(400^2 - 4(-5)(-3,997,500))) / (2(-5))

Simplifying further:
x = (-400 ± sqrt(160,000 - 79,950,000)) / (-10)
x = (-400 ± sqrt(-79,790,000)) / (-10)

Since the discriminant (b^2 - 4ac) is negative, there are no real solutions. Therefore, there is no x-intercept.

Step 3: Determine the sign of the quadratic function in the intervals to find the range of x.

Based on the coefficient of the x^2 term (-5), the quadratic function (-5x^2 + 400x - 3,997,500) opens downwards. Therefore, the function is negative for values of x outside the interval where the vertex is located.

Step 4: Calculate the x-coordinate of the vertex using the formula:
x = -b / (2a)

For our equation, with a = -5 and b = 400, we have:
x = -400 / (2(-5))
x = -400 / (-10)
x = 40

The x-coordinate of the vertex is 40.

Step 5: Determine the sign of the quadratic function in the interval around the vertex.

Test a value less than 40, such as x = 30:
P(30) = -5(30)^2 + 400(30) + 2500
P(30) = -5(900) + 12,000 + 2500
P(30) = -4500 + 12,000 + 2500
P(30) = 18,000

Since P(30) = 18,000 > 0, the function is positive.

Test a value greater than 40, such as x = 50:
P(50) = -5(50)^2 + 400(50) + 2500
P(50) = -5(2500) + 20,000 + 2500
P(50) = -12,500 + 20,000 + 2500
P(50) = 18,000

Since P(50) = 18,000 > 0, the function is positive.

Step 6: Write the range of x-values where P(x) is greater than or equal to zero.

Since the quadratic function is always positive for all x, the solution to the inequality is the set of all real numbers.

Therefore, any amount spent on advertising will result in a profit of at least $4,000,000.

Answer 4

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.

First, let's rewrite the inequality with the given profit function P(x):
-5x^2 + 400x + 2500 ≥ 4,000.

Now, let's solve the inequality step by step:

Step 1: Subtract 4,000,000 from both sides to isolate the quadratic expression:
-5x^2 + 400x + 2500 - 4,000,000 ≥ 0.

Step 2: Simplify the expression:
-5x^2 + 400x - 3,997,500 ≥ 0.

Step 3: Divide the entire inequality by -1 to make the leading coefficient positive:
5x^2 - 400x + 3,997,500 ≤ 0.

Now, we need to find the values of x that satisfy this inequality. To do that, we can find the solutions to the equation 5x^2 - 400x + 3,997,500 = 0 and then determine the range of x that satisfies the inequality.

To solve the equation, we can use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a).

For the equation 5x^2 - 400x + 3,997,500 = 0:
a = 5, b = -400, and c = 3,997,500.

Plugging these values into the quadratic formula:
x = (-(-400) ± sqrt((-400)^2 - 4(5)(3,997,500))) / (2(5)).

Simplifying this expression further:
x = (400 ± sqrt(160,000 + 79,950,000)) / 10.

x = (400 ± sqrt(80,110,000)) / 10.

x = (400 ± sqrt(8,011,000)) / 10.

Now, we can calculate the values of x by taking both the positive and negative square root of 8,011,000:

x = (400 + sqrt(8,011,000)) / 10,

and

x = (400 - sqrt(8,011,000)) / 10.

These two values will be our solutions for the equation 5x^2 - 400x + 3,997,500 = 0. Now, we need to determine which range of x satisfies the inequality 5x^2 - 400x + 3,997,500 ≤ 0.

To do that, we need to find the critical points. The critical points occur when the quadratic expression is equal to zero. Therefore, we need to solve the equation 5x^2 - 400x + 3,997,500 = 0.

By using the quadratic formula as shown before, we find the two critical points:
x = (400 + sqrt(8,011,000)) / 10 ≈ 510.18,

and

x = (400 - sqrt(8,011,000)) / 10 ≈ 78.82.

Now, we can determine the range of x that satisfies the inequality based on the critical points and the nature of the parabola.

Since the coefficient of x^2 is positive (+5), the parabola opens upwards. Therefore, the values of x that satisfy the inequality will be between the two critical points.

Finally, the amount that must be spent on advertising to obtain a profit of at least $4,000,000 lies between approximately $78,820 and $510,180 (in thousands of dollars).