# Tutor profile: Ricardo U.

## Questions

### Subject: Basic Math

You are father of three, and you forget things from time to time. One day, you forgot the age of your childrens so you asked them how old they were. They were not happy with that. The son #1 then replied: "In 15 years, the sum of our ages will be twice your current age" You did know you were 40 years old at that time. However you could not answer the question. At the end, you remembered that the kids are triplets! So, how old were they when you asked them?

First, let us name the variables. X: age of the son #1 at the moment they were asked about their age Y: age of the son #2 at the moment they were asked about their age Z: age of the son #3 at the moment they were asked about their age A: age of the father at the moment he asked his childrens about their age Then: (X + 15) + (Y + 15) + (Z + 15) = 2*A ---> "In 15 years, the sum of our ages will be twice your current age." X + Y + Z + 45 = 2*A And A = 40 ---> "You did know you were 40 years old at that time." X + Y + Z + 45 = 80 Since the kids are triplets: X = Y = Z (with difference in some minutes or hours). Then, these variables can be added to yield: 3*X + 45 = 80 And solving for X: X = (80 - 45)/3 = 11.67 So your sons were 11.67 years, or approximately 11 years and 8 months old, when you asked about their age.

### Subject: Calculus

You and your friends are playing in a vacuum chamber throwing rocks (you are equipped with a nice oxygen system to keep breathing). Since you and your friends were studying basic kinematics, you want to know the following: if two of you throw a rock in the same direction, what conditions should be held so that the maximum altitude of each rock is attained at the same time?

From your physics book, you know the formulae of parabolic movement. Since you are in a vacuum chamber, air resistance is neglected. The formulae is: h(t) = Vo*t*sin(a) - (1/2)*g*t^2; Where t= time; h = height of the rock at time t; Vo = initial velocity; a: launch angle; g: gravitational acceleration. Since a maximum height is wanted, one should do: dh(t)/dt = 0 If that is performed on the kinematic expression, we then have: dh(t)/dt = Vo*sin(a) - g*t (please not that Vo and sin(a) are constant values for a given launch. g is always a constant). 0 = Vo*sin(a)-g*t So the needed time for the rock to reach its maximum height is t = Vo*sin(a)/g Therefore, for two rocks to achieve their maximum height at the same time, their Vo*sin(a) product should be the same. Note that absolute values of Vo and sin(a) are not relevant, just the value of their product!

### Subject: Algebra

Imagine you have twin brothers and that you are rich. Both brothers ask you, for separate, to lend them some money. You are not sure if they will pay you back, so you have to find a way to see if you can trust them. You asked them for a mathematical formulae that allows you to estimate the time it will take each of them to pay the loan. You then have a small conversation with each twin while the other awaits outside the room. You ask the first tween for the formulae and he hands you the following expression: (t-sqrt(a))(t+sqrt(a)) = 0 "Just solve for the time t. It will depend on how much money you are willing to lend me, that's the a". You then ask the second tween for the formulae. He hands you exactly the same thing: (t-sqrt(a))(t+sqrt(a)) = 0 "Just solve for the time t. It will depend on how much money you are willing to lend me, that's the a". What question would you ask to the second twin that might allow you to conclude if they are reliable?

The question would be: if I lend same amount of money to you and your brother, would you pay at the same time? There are three possible answers to this: a) Yes, we will pay at the same time. b) I have no idea! c) No, we will not pay you at the same time. Since the formulae both twins handed you is a quadratic equation, it has two solutions: t = sqrt(a) and -t = sqrt(a). Since t represents the time, t > 0; a is money, hence a >0. Therefore, -t = sqrt(a) is not a valid solution. Answer (b) suggests that some of them could pay you in a negative time, i.e., never paying you back. The answer (c) confirms that one of them is planning to pay you in a negative time, that is, never paying you. Therefore, with this question, you could be more confident if the second twin answers the option (a).

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