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# Tutor profile: Ryan C.

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Ryan C.
Certified Math and Science Teacher
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## Questions

### Subject:Physics (Waves and Optics)

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Question:

A 2.5 cm tall object is placed 6.0 cm in front of a thin convex lens with a focal length of 8.0 cm. What will be the characteristics of the resulting image?

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Ryan C.

In order to answer this question, we first need to figure out where the image will appear. To do this, we'll use the thin lens equation: 1/do + 1/di = 1/f f is the focal length and do and di are the object distance and image distance respectively (sometimes p and q are used instead, but I prefer do and di because they are clearer.) Our focal length is 8.0 cm and since it is a convex lens, f will be positive (if it were concave, it would be negative). The object distance is how far away our object is from the lens, which is 6.0 cm. The object distance is always positive. The image distance will be how far the image is from the lens, which is what we want to know right now. So we have: do = 6.0 cm di = we want to know f = 8.0 cm Plugging these into our equation, we get: 1/6.0 + 1/di = 1/8.0 Subtract 1/6.0 from both sides to get: 1/di = -0.042 I rounded here to write the number down, but you're better off leaving the whole thing in your calculator before doing the next part. We currently have 1/di, but we want di. To get that, we need to raise both sides to the negative first power, which essentially flips the numerator and denominator. Most calculators have a x^-1 button you can use to do this. Doing so should result in: di = -24 The fact that our image distance is negative reveals the first characteristic. Real images have positive image distances and virtual images have negative ones. So our image is going to be virtual. For a lens, this means the image will be on the same side of the lens as the object and since it's not a real image, we would not be able to project it on a screen, since light is not actually going there; it only appears to be. For the last two characteristics, we're going to need a second equation: the magnification equation. This one looks like: M = hi/ho = -di/do M is the magnification, which is defined as the ratio of the image height to the object height (hi and ho). This is conveniently equal to the ratio of the image distance to the object distance (with a negative sign thrown in). We could use our object height of 2.5 cm here to find out how tall our image is going to be, but we can answer the question about the characteristics with just the magnification. Plugging into the magnification equation gives us: M = -(-24)/6.0 The two negatives cancel and make a positive and leave us with: M = 4.0 This magnification tells us a few things. The fact that it is positive means the image is upright. If it were negative, the image would be flipped upside down, or "inverted." Additionally, the magnification has an absolute value greater than 1. This means the image is bigger than the object (4 times bigger, in fact). We can say our image is "enlarged" rather than "diminished." We can now answer the question by saying the image will be VIRTUAL, UPRIGHT, and ENLARGED. Before we go, let's do a quick sanity check on our answer. A magnifying glass is an optical instrument made from a convex lens. When we put something close to a magnifying glass (within its focal length) we obviously would in fact expect an enlarged image (that's kind of the point of them). We would also expect an upright image (it would be pretty annoying if magnifying glasses flipped things). Finally, magnifying glasses produce virtual images, because we could not put a sheet of paper where the image appears to be and see it projected on the paper as a real image would be. So we can be confident that our answer of virtual, upright, enlarged matches what we would expect from reality.

### Subject:Physics

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Question:

If a ball is dropped from a height of 22 meters, how long will it take to reach the ground? Assume you are on Earth and can ignore air resistance.

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Ryan C.

### Subject:Algebra

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Question:

42 people went to the movies. The movie theater sold \$300 worth of tickets. An adult ticket costs \$10 and a children's ticket costs \$6. How many children's tickets were sold? How many adult tickets were sold?

Inactive
Ryan C.

Given a word problem like this, where we have two things we don't know (a.k.a. variables), what we want to do is set up a system of equations. As long as we have as many unique equations as we have variables, we can solve for each of those variables. One thing we know is that 42 people went to the movies. Assuming all of those people qualify as adults or children (at least when it comes to buying tickets) this gives us the equation: A + C = 42 where A represents the number of adult tickets sold and C represents the number of children's tickets sold. For our second equation, we can look at the money. The total amount of money made is going to be the adult price times the number of adult tickets plus the children's price times the number of children's tickets. In other words: 10A + 6C = 300 Now that we have our system of two equations, we have a couple ways we can solve them. These are known as the substitution and elimination methods. I'm going to use the substitution method here, since it's slightly more convenient for this problem, but either would work. First we'll take our first equation and subtract C from both sides. This is because when solving an algebra problem, you want to do the opposite operation from what was being done in order to isolate your variable (C was being added to A, so we're subtracting it) and whatever we do, we have to do it from both sides - otherwise the equation is no longer balanced and therefore untrue. This gives us: A = 42 - C Now we have an expression for A that we can substitute into the second equation any time we see A there. The benefit of this is that we now have an equation with just one variable and those are much easier to solve. Our new equation is: 10(42-C) + 6C = 300 Remembering to use the distributive property, we can simplify this to: 420 -10C + 6C = 300 Then we can combine like terms to get: 420 - 4C = 300 Then subtract 420 from both sides: -4C = -120 And finally divide both sides by negative 4 to solve for C: C = 30 This means 30 children's tickets were sold and we've answered the first half of the question. Fortunately, the second half will be much faster now that we have that information. We can simply use the fact that C = 30 and plug that into our rearranged first equation from earlier, giving us: A = 42 -30 = 12 This means 12 adult tickets were sold and we've completely solved the problem.

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