Question

The number n of people using the elevator in an office building every hour is given by n = t2 − 10t + 40. In this equation, t is the number of hours after the building opens in the morning, 0 ≤ t ≤ 12. Will the number of people using the elevator ever be less than 15 in any one hour? Use the discriminant to answer.

Answer 1

The number of people using the elevator will never be less than 15 in any one hour according to the given equation n = t^2 - 10t + 40.

Step-by-step explanation:

The number of people using the elevator in an office building every hour is given by the equation n = t2 - 10t + 40, where t is the number of hours after the building opens in the morning, and 0 ≤ t ≤ 12. To determine if the number of people using the elevator will ever be less than 15 in any one hour, we can use the discriminant of the equation. The discriminant, denoted as Δ, is calculated as b² - 4ac, where a, b, and c are the coefficients of the quadratic equation. In this case, a = 1, b = -10, and c = 40. Evaluating the discriminant, we get Δ = (-10)² - 4(1)(40) = 100 - 160 = -60.

Since the discriminant is negative (Δ < 0), it means the quadratic equation has no real roots. Therefore, the number of people using the elevator will never be less than 15 in any one hour.

Answer 2

The number of persons using the elevator at any hour is never going to be less that 15.

Step-by-step explanation:

To solves this you have to suppose that there are at least 15 persons on the elevator, and the equation is converted into an inequation:

`t^2-10t+40<15\\t^2-10t+25<0`

Now you transform the inequation back to an equation to solve it:

`t^2-10t+25=0`

You need to know if there is any negative solution for the equation, to do this you can use the discriminant for a quadratic equation:

`D=b^2-4ac`

In this case, you have a=1, b=-10, c=25

`D= (-10)^2-4(1)(25)\\D=0`

Since the discriminant is 0 and a<0 the equation always is going to be positive. Therefore, the number of persons using the elevator at any hour is never going to be less than 15.