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Justin S.
Student at Princeton University, Tutor for Four Years
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Trigonometry
TutorMe
Question:

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?

Justin S.

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.

Geometry
TutorMe
Question:

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.)

Justin S.

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.

Algebra
TutorMe
Question:

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.)

Justin S.