Question

The profit function p(x) of a tour operator is modeled by p(x) = −2x^2 + 700x − 10000, where x is the average number of tours he arranges per day. What is the range of the average number of tours he must arrange per day to earn a monthly profit of at least $50,000?

Answer 1

Range of the average number of tours is between 150 and 200 including 150 and 200.

Explanation:

Given:

The profit function is modeled as:

`p(x)=-2x^2+700x-10000`

The profit is at least $50,000.

So, as per question:

`p(x)\geq50000\\-2x^2 + 700x-10000\geq 50000\\-2x^2+700x-10000-50000\geq 0\\-2x^2+700x-60000\geq 0\\\\\textrm{Dividing by 2 on both sides, we get}\\\\-x^2+350x-30000\geq 0`

Now, rewriting the above inequality in terms of its factors, we get:

`-1(x-150)(x-200)\geq 0\\(x-150)(x-200)\leq 0`

Now,

`x<150,(x-150)(x-200)>0\\x>200,(x-150)(x-200)>0\\For\ 150\leq x\leq200,(x-150)(x-200)\leq 0\\\therefore x=[150,200]`

Therefore, the range of the average number of tours he must arrange per day to earn a monthly profit of at least $50,000 is between 150 and 200 including 150 and 200.