Answer:
Range of the average number of tours is between 150 and 200 including 150 and 200.
Step-by-step explanation:
Given:
The profit function is modeled as:
[tex]p(x)=-2x^2+700x-10000[/tex]
The profit is at least $50,000.
So, as per question:
[tex]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[/tex]
Now, rewriting the above inequality in terms of its factors, we get:
[tex]-1(x-150)(x-200)\geq 0\\(x-150)(x-200)\leq 0[/tex]
Now,
[tex]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][/tex]
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.
