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)=-x^{2}+35x - 200 for 0 ≤ x ≤ 45. how many bottles of tonic must be sold to make at least $50 in profit? write the largest interval containing all possible answers.

Explanation:

Step1: Set up the profit - inequality

We want $P(x)\geq50$, where $P(x)=-x^{2}+35x - 200$ and $0\leq x\leq45$. So, we set up the inequality $-x^{2}+35x - 200\geq50$.

Step2: Rearrange the inequality

Move all terms to one side to get a quadratic - inequality in standard form: $-x^{2}+35x - 200 - 50\geq0$, which simplifies to $-x^{2}+35x - 250\geq0$. Multiply through by - 1 (and reverse the inequality sign) to obtain $x^{2}-35x + 250\leq0$.

Step3: Solve the quadratic equation

First, solve the corresponding quadratic equation $x^{2}-35x + 250 = 0$. Use the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$, where $a = 1$, $b=-35$, and $c = 250$.
$x=\frac{35\pm\sqrt{(-35)^{2}-4\times1\times250}}{2\times1}=\frac{35\pm\sqrt{1225 - 1000}}{2}=\frac{35\pm\sqrt{225}}{2}=\frac{35\pm15}{2}$.
The solutions are $x_1=\frac{35 + 15}{2}=\frac{50}{2}=25$ and $x_2=\frac{35 - 15}{2}=\frac{20}{2}=10$.

Step4: Determine the solution interval

The quadratic function $y=x^{2}-35x + 250$ is a parabola that opens upward (since $a = 1>0$). The inequality $x^{2}-35x + 250\leq0$ is satisfied when $10\leq x\leq25$.
We are interested in the largest value of $x$ in the interval $[0,45]$ that satisfies the inequality.

Answer:

25