Tutor profile: Sami A.

How many ways can 8 people sit around a circular table?

Begin without taking into account the circular table. Recognize that the number of orderings of n distinct objects is n!. Therefore, the number of ways to order 8 people is 8!. Now, introduce the round table. Notice that certain orderings would yield identical configurations on a round table. For example the orders 12345678 and 23456781 code to the same table setup. Because, for a given setup, any of the 8 seats could be taken as the beginning of the ordering, you must divide the number of orders, 8! by 8 to get a final answer of 8!/8 = 7!.

How many real roots does the equation $$-3 = 5x^2 -6x$$ have?

First, recall the quadratic formula. Recognize that the number of real roots depends on the sign of the determinant, $$b^2 - 4ac$$. If this value is positive, there are 2 real roots, if it is negative, there are not real roots, and if it is 0, there is 1 double root. Calculate this value as $$ (-6)^2 - 4*5*3 = -24$$. Because this is negative, you know this equation has no real roots.

With what velocity must I throw a ball upwards in order for it to remain in the air for 10 seconds?

Recognize this problem as an application of kinematic equations (a set of equations derived from Newton's laws about how objects move). Now gather the information the problem is giving you. You are given, explicitly, the total air time as 10 seconds. You are also implicitly given an acceleration, g, of -9.8 m/s^2 and a total displacement of 0 (the ball starts and ends on the ground). Now choose an equation using these known values which includes the value you are looking for: initial velocity. We will select $$x = x_0 + v_0t + .5a t^2$$. Now plug in our known values, using that $$x = x_0$$ to get $$0 = v_0*10 - .5*9.8* 100$$. Solve for $$v_0$$ to get 49 m/s. That's a really hard throw!!