Question

If there are n people in a room, the number n of possible handshakes by all the people in the room is given by the function n(n) = n(n – 1)/2. for what number n of people is 66 ≤ n ≤ 300?

Answer 1

To find the number of people for which the possible handshakes are between 66 and 300, we solve the inequalities generated by doubling the original equation and then use quadratic equations to find the range for n.

Step-by-step explanation:

The function n(n) = n(n – 1)/2 gives the number of handshakes possible between n people. To find the number of people for which there are at least 66 but no more than 300 handshakes, we need to find the values of n that satisfy 66 ≤ n(n – 1)/2 ≤ 300.

First, we would double each side of the inequalities to get rid of the fraction: 132 ≤ n(n – 1) ≤ 600. Next, we solve for n by finding the values that satisfy these inequalities through a process of trial and error or by using the quadratic formula after rearranging the inequality into a standard quadratic equation form: n² - n - 132 ≤ 0 and n² - n - 600 ≤ 0.

After finding the roots of the quadratic equations, we take the integer values of n that lie between the roots and satisfy both inequalities. This will give us the range of n for which the number of handshakes is between 66 and 300.