Your class is selling boxes 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 equals negative 0.5 x squared plus 36 x minus 134 models the profit of the fundraiser. What's the smallest amount, in dollars, that you can charge and make a profit of at least $388?.

Question

Your class is selling boxes 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 equals negative 0.5 x squared plus 36 x minus 134 models the profit of the fundraiser. What's the smallest amount, in dollars, that you can charge and make a profit of at least $388?.

Answer 1

gee, why all those words?

p(x) = -0.5x^2 + 36x - 134
So, just solve
-0.5x^2 + 36x - 134 = 388
As with all quadratics, there are two solutions. Pick the one you need. Be sure to round up.

Answer 2

To find the smallest amount, in dollars, that you can charge and make a profit of at least $388, we need to solve the equation p ≥ $388.

The equation given for profit is p = -0.5x^2 + 36x - 134.

Substituting p = $388 into the equation, we get:

$388 = -0.5x^2 + 36x - 134

Rearranging the equation to standard form, we have:

0.5x^2 - 36x + 522 = 0

To solve this quadratic equation, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 0.5, b = -36, and c = 522.

Plugging these values into the quadratic formula, we have:

x = (-(-36) ± √((-36)^2 - 4(0.5)(522))) / (2(0.5))

Simplifying this equation, we get:

x = (36 ± √(1296 - 1044)) / 1

x = (36 ± √252) / 1

x = (36 ± 15.87)

So, the two possible values for x are:

x₁ = 36 + 15.87 ≈ 51.87

x₂ = 36 - 15.87 ≈ 20.13

Since we want to find the smallest amount that can be charged, we would choose x₂ ≈ 20.13.

Therefore, the smallest amount, in dollars, that you can charge and make a profit of at least $388 is approximately $20.13.

Answer 3

To find the smallest amount, in dollars, that you can charge and make a profit of at least $388, we need to solve the equation p ≥ $388.

The profit equation is given by p = -0.5x^2 + 36x - 134.

So, we need to solve the inequality -0.5x^2 + 36x - 134 ≥ 388.

First, let's rewrite the inequality to make it easier to solve: -0.5x^2 + 36x - 134 - 388 ≥ 0.

Simplifying the left side of the inequality: -0.5x^2 + 36x - 522 ≥ 0.

To solve this quadratic inequality, we can use different methods like factoring, completing the square, or using the quadratic formula. In this case, factoring is not straightforward, so we'll use the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / 2a.

For our quadratic equation -0.5x^2 + 36x - 522 = 0, the coefficients are:
a = -0.5, b = 36, and c = -522.

Substituting these values into the quadratic formula, we get:

x = (-36 ± sqrt(36^2 - 4*(-0.5)*(-522))) / (2*(-0.5)).

Simplifying the expression:

x = (-36 ± sqrt(1296 - (-1044))) / (-1).

x = (-36 ± sqrt(2340)) / (-1).

x = (-36 ± sqrt(4 * 585)) / (-1).

x = (-36 ± 2sqrt(585)) / (-1).

Now, we have two possible solutions: x = (-36 + 2sqrt(585)) and x = (-36 - 2sqrt(585)).

To find the smallest amount, we need to evaluate these solutions. Let's calculate both:

x = (-36 + 2sqrt(585)) ≈ 25.867.

x = (-36 - 2sqrt(585)) ≈ -62.867.

Since selling a box of seeds at a negative price doesn't make sense in this context, we can conclude that the smallest amount, in dollars, that you can charge to make a profit of at least $388 is approximately $25.867.