An air traffic controller (ATC) is sitting in the top of a 160 m high tower. He spots a plane on the tarmac ahead of him. The ATC must calculate how far away the plane is from the base of the tower he is in: does he have enough information to do this?
Answer: No, he does not. Here's why: It's easiest to see this explanation with the aid of a visual drawing, but I'll try my best here: OOOOO - - - OOOOO - - - lll - - - 160 m lll - - - lll - - - lll - - - lll - - - lll___________________________________PLANE___ X (So this is a very basic drawing and isn't accurate at all whatsoever.) (The "- - -" symbols are dashed lines representing the controller's line of sight.) Essentially, the ground, tower, and ATC's line of sight all make one big triangle. The distance that the ATC needs to calculate is marked by "X." We cannot calculate X because we only have one angle in the triangle, the 90-degree angle made by the tower and ground. We cannot use 'sin' and 'cos' functions because we don't have a length for the hypotenuse, and the 'tan' function is unavailable as well since the tangent function cannot be used with a 90-degree angle in a right triangle. Therefore, the ATC cannot calculate how far away the plane is from the base of his tower.
A soda can company is looking to remodel its cans. Currently, the cans are perfect cylinders and have a 3 cm radius and a 12 cm height. The new cans will have a radius of 4 cm. How tall must the cans be if the company wishes to make no changes to the volume of the cans themselves? (For the sake of simplicity, the value of pi can be shortened to 3.14.)
Well the first thing to do here is to calculate the original volume since we don't want to change that at all. The formula to calculate the volume of a cylinder is "Volume = pi*r^2*h." By substituting 3 for "r," 12 for "h," and 3.14 for "pi," we get: Volume = 3.14 * 3^2 * 12 This equals 339.12 cubic centimeters. Now, we know both the volume and radius of the new cans. What we don't know is the height, which we will now solve for. Using the same formula as before, 339.12 = 3.14 * 4^2 * h Simplifying just the right side of the equation give us: 339.12 = 50.24 * h Dividing both sides by 50.24 give us: h = 6.75 Therefore, the new height of the cans must be 6.75 cm to ensure that the volume is consistent. When doing these types of problems, it is very helpful to make visual diagrams so that you can visualize it and so that your variable/numbers don't get mixed up and/or confused.
Jimmy's art class and some teachers are all attending an art museum to see a Van Gogh exhibit. If adult tickets are priced at $10 and student tickets are priced at $5, how many students are in Jimmy's class besides Jimmy? (A total of 20 people attend the exhibit, spending a total of $120.)
Right off the bat, we can see that this is a problem dealing with systems of equations. In order to solve this problem, the first thing we must do is define/identify the variables we are dealing with. The question is asking for how many students attend the exhibit besides Jimmy, so we'll make a variable for that as "S." However, we must also account for the teachers, so we'll call the number of teachers as "T." In a system of equations problem, there are always two equations that we must solve for. One equation deals with the amount of people that attended the exhibit (20) and another is based on the total amount of money spent ($120). We know that 20 people go, so the total amount of students and teachers combined should be 20. In other words, S + T = 20 We also know that adult tickets cost $10 and student tickets are $5. Furthermore, we know that the total spent is $120. Therefore, the sum of the respective ticket costs multiplied by the respective amounts of attendees should equal $120. In other words, 5S + 10T = 120 The reason why it equals 120 and not 20 is because when you deal with money on the left side of an equation, you must also account for that on the right side. Now we have these two equations to solve. S + T = 20 5S + 10T = 120 There are two main ways to solve this problem: solving by cancellation, and substitution. ***Method 1: Cancellation*** S + T = 20 5S + 10T = 120 In cancellation, we are getting rid of a variable so that we can solve an equation with one variable. It doesn't matter which variable is cancelled out, because you can always go back and substitute to solve again. However in this case, we want to get rid of the "T" variable. This is because if you go back and read the original question, you'll see that the question doesn't ask for the amount of teachers attending at all. Therefore, we can take that variable out. But how exactly would we do that? Well, the whole point of cancellation is to multiply the equations themselves in such a manner so that adding the two equations cancels out one variable (and not both). We can do that, in this case, by multiplying everything in the first equation by "-10." -10(S + T = 20) Multiply the first equation by "-10." 5S + 10T = 120 Simplify. -10S - 10T = -200 5S + 10T = 120 Upon adding the two above equations together, we get: -5S = -80 Notice how the "T" variable is now gone! Simplifying gets us: S = 16 Therefore, 16 students are in the class. BUT WE'RE NOT DONE! The question asks "how many students are in Jimmy's class besides Jimmy?" So, the answer is: 15 students are in Jimmy's class besides Jimmy. (Always remember to answer word problems with words!) ***Method 2: Substitution*** In substitution, we are substituting one variable with another, hence the name. S + T = 20 Remember these two equations? 5S + 10T = 120 Just like before, we want to get rid of the "T" variable ASAP since solving for it simply wastes time. In substitution, we always want to place the variable we want gone by itself on one side of the equal sign. In other words, we re-arrange the first equation from S + T = 20 to T = 20 - S Then, we substitute this new first equation into the second one as such: 5S + 10(20 - S) = 120 Don't be scared if this looks different, all we did was replace "T" with "20 - S." Simplifying that equation gives us: 5S + 200 - 10S = 120 Further simplification shows: -5S = -80 Therefore, S = 16. Just as before, since the question asks "how many students are in Jimmy's class besides Jimmy," the answer is: 15 students are in Jimmy's class besides Jimmy. Notice how two different methods got the same correct answer, choose whichever one works better for you.