Question

(5 points) An outdoor adventure company advertises that they will provide a guided

mountain bike trip and a picnic lunch for $50 per person. They must have a guarantee

of 30 people to do the trip. Furthermore, they agree that for each person in excess of

30, they will reduce the price per person for all riders by $0.50. How many people will

it take to maximize the company's revenue?

Answer 1

It take 65 people to maximize the revenue.

Explanation:

Consider the provided information.

Let x is the number of people who take part in trip.

They charge $50 per person and they will reduce the price per person for all riders by $0.50 for each person excess of 30.

The price for each person is
`50-0.50(x-30)` where x is greater or equal to 30.

`50-0.50x+15`

`-0.50x+65`

Now write a revenue function by multiplying the number of people with the price per person.

`f(x)=x(-0.50x+65)`

`f(x)=-0.50x^2+65x`

We need to maximize the company's revenue.

The above function is a parabola that opens downward because the coefficient of x²is negative.

Therefore, the maximum of the function is at its vertex.

If the equation of the parabola is
`f(x)=ax^2+bx+c` then we can find the coordinate of vertex at
`((-b)/(2a),(4ac-b^2)/(4a))`

Calculate the value of
`(-b)/(2a)` for the function
`f(x)=-0.50x^2+65x`.

`(-65)/(2(-0.50))=65`

Therefore, it take 65 people to maximize the revenue.