Two trains leave towns 1190 km apart at the same time and travel toward each other. One train travels 14 km/h slower than the other. If they meet in 5 hours, what is the rate of each train?
Let x be the rate of the slower train and let y be the rate of the faster train. Using the formula distance = rate * time, we have the following information. Slower train: * x km/h (rate) * 5 hours (time) * 5x km (distance) Faster train: * y km/h (rate) * 5 hours (time) * 5y km (distance) The trains begin 1190 km apart, travel toward each other, and eventually meet. So, 1190km is the sum of their distances traveled. We get the following equation: 5x + 5y = 1190. The slower train travels 14 km/h slower than the faster train. From this, we get another equation: x = y - 14. This gives us a system of equations. Using the equation we just created, we can substitute for x in the original equation: 5(y - 14) + 5y = 1190. Once the algebra is complete and y is solved for, you should get y = 126. Now that you have y, plug in y = 126 into x = y - 14 to get x = 112. So you have x = 112 and y = 126. The rate of the slower train is 112 km/h and the rate of the faster train is 126 km/h.
If you are standing 100 feet away from the Eiffel Tower. The line from your feet to the top of the Tower and the line that represents the ground give an angle of 84.6 degrees. What is the approximate height of the Eiffel Tower?
To obtain the height of the Eiffel Tower, you must use your trigonometric ratios. Since you are given the distance (adjacent) and the angle and we are looking for the height (opposite), then it would be most appropriate to use tan(theta) = opposite/adjacent. Now we have to fill in the blanks with what we know from the given information. We plug in tan(84.6) = opposite/100. We leave opposite there since that is what we are looking for. Now to use algebra to solve for "opposite." Multiply both side by 100 to get tan(84.6) * 100 = opposite. Plug in tan(84.6) * 100 to your calculator to obtain your result. Make sure your calculator is in degrees since the question was stated in degrees. Your answer should be approximately 1058 (rounded to the nearest whole number).
If you were to high-five each person in a room of five people (including yourself), how many high-fives occurred?
First, let's think about how many people YOU gave a high-five to. You gave out four high-fives. So, that means everyone gave our four high-fives, right? Well, let's see. You gave Person 1, Person 2, Person 3, and Person 4 a high-five. Now let's look at Person 1. He gave you, Person 2, Person 3, and Person 4 a high-five. But wait - isn't you high-fiving Person 1 the same thing as Person 1 high-fiving you? So we have to be sure to not double count that. So Person 1 gave out three unique high-fives from your own. If we follow this pattern with each person in the room, you will get 4+3+2+1 = 10 high-fives that occurred. This is a non-linear pattern.