Question

the profit, in dollars, made by selling x bottles of 100% all - natural certified free - trade organic sasquatch tonic is given by p(x)=-x² + 40x - 150 for 0≤x≤50. how many bottles of tonic must be sold to make at least $150 in profit? write the largest interval containing all possible answers.

Explanation:

Step1: Set up the profit - inequality

We want $P(x)\geq150$, where $P(x)=-x^{2}+40x - 150$. So, we set up the inequality $-x^{2}+40x - 150\geq150$.

Step2: Rearrange the inequality

Move all terms to one side to get a quadratic inequality: $-x^{2}+40x - 300\geq0$. Multiply through by - 1 (and reverse the inequality sign) to obtain $x^{2}-40x + 300\leq0$.

Step3: Solve the corresponding quadratic equation

Set $y=x^{2}-40x + 300$ and solve $x^{2}-40x + 300 = 0$. Using the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ for the quadratic equation $ax^{2}+bx + c = 0$. Here, $a = 1$, $b=-40$, and $c = 300$. Then $x=\frac{40\pm\sqrt{(-40)^{2}-4\times1\times300}}{2\times1}=\frac{40\pm\sqrt{1600 - 1200}}{2}=\frac{40\pm\sqrt{400}}{2}=\frac{40\pm20}{2}$. The roots are $x_1=\frac{40 + 20}{2}=30$ and $x_2=\frac{40-20}{2}=10$.

Step4: Determine the solution interval

The quadratic function $y=x^{2}-40x + 300$ is a parabola opening upwards (since $a = 1>0$). The inequality $x^{2}-40x + 300\leq0$ is satisfied when $10\leq x\leq30$.

Answer:

The number of bottles $x$ must satisfy the interval $[10,30]$. So, the largest interval containing all possible answers is $[10,30]$.