Question

The profit function p(x) of a tour operator is modeled by p(x) = −2x2 + 900x − 40000, 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 $60,000?
between 200 and 250; exclusive
between 200 and 250; inclusive
less than 250
more than 200

Answer 1

Solving the quadratic function, it is found that the range is of:

between 200 and 250; inclusive

The profit for x tours is modeled by the following equation:

$p(x) = -2x^2 + 900x - 40000$

It is concave down, so the monthly profit will be of at least $60,000 between the two roots when $p(x) = 60000$. Thus:

$60000 = -2x^2 + 900x - 40000$

$2x^2 - 900x + 10000 = 0$

Simplifying by 2:

$x^2 - 450x + 50000 = 0$

Which has $a = 1, b = -450, c = 50000$

Then:

$\Delta = (-450)^2 - 4(1)(50000) = 2500$

$x_{1} = \frac{-(-450) + \sqrt{2500}}{2} = 250$

$x_{2} = \frac{-(-450) - \sqrt{2500}}{2} = 200$  

At least $60,000 includes $60,000, thus the interval is inclusive, and the correct option is:

between 200 and 250; inclusive

A similar problem is given at https://brainly.com/question/25181401