Question

Your class is selling boxs of flower seeds as a fundraiser The total profit p depends on the amount x that your class charges for each box of seeds. The equation p= -0.5x^2+36x-201 models the profit of the fundraiser.​ What's the smallest​ amount, in​ dollars, that you can charge and make a profit of at least ​$​393?

Answer 1

To find the smallest amount that you can charge and make a profit of at least $393, you need to solve a quadratic inequality. The smallest amount you can charge is approximately $36.19.

Step-by-step explanation:

To find the smallest amount that you can charge and make a profit of at least $393, you need to set up an inequality. The inequality is:

p ≥ 393

Substituting the equation for profit, -0.5x^2 + 36x - 201, into the inequality, we have:

-0.5x^2 + 36x - 201 ≥ 393

Next, rearrange the inequality to solve for x:

-0.5x^2 + 36x - 201 - 393 ≥ 0

-0.5x^2 + 36x - 594 ≥ 0

To solve this quadratic inequality, you can use different methods such as graphing, factoring, or using the quadratic formula. In this case, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

Plugging in the values from the quadratic inequality, we get:

x = (-36 ± √(36^2 - 4(-0.5)(-594))) / (2(-0.5))

Calculating the values inside the square root gives:

x = (-36 ± √(1296 - 1188)) / (-1)

x = (-36 ± √108) / (-1)

Simplifying further gives:

x = -18 ± 3√3

Since we're interested in the smallest amount, we take the smaller value:

x = -18 - 3√3

Therefore, the smallest amount you can charge and make a profit of at least $393 is approximately $36.19.