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Andrew B.

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Basic Chemistry

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Question:

Balance the following equation: Fe + Cl2 = FeCl3

Andrew B.

Answer:

To have a balanced equation, there must be the same quantity of each element on both sides of the equation. In general, it is best to start with the elements that require manipulation on both sides. In this case, since there are two atoms of Cl in the reagents, and three atoms in the product, we will begin there. We determine the lowest common denominator of the numbers, which is six. Our first step is then: Fe + 3Cl2 = 2FeCl3 Now, we have the same amount of Cl on both sides. However, we still have differing amounts of Fe. Since one is a factor of two, we can multiply the reagent Fe to finish balancing this equation. 2Fe + 3Cl2 = 2FeCl3. We now have two atoms of Fe on both sides of the equation, and six atoms of Cl on both sides. The equation is balanced.

Calculus

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Question:

Differentiate y = 5sin(x) - 2cos(x).

Andrew B.

Answer:

In solving this problem, we must use the rules for the differentiation of trigonometric functions. The applicable ones are: D{sin(x)} = cos(x), and D{cos(x)} = -sin(x). Also, recall that when the function we seek to differentiate is multiplied by a constant, we can "move" the constant outside of the differentiation. That is, D{cf(x)} = c*D{f(x)}. Now, we are able to solve the problem. We begin by simply using the notation of differentiation. y' = D{5sin(x)} - D{2cos(x)} Next, we can move the constants in the equation. y' = 5D{sin(x)} - 2D{cos(x)} Then, we apply the rules for trigonometric differentiation. y' = 5cos(x) - 2(-sin(x)) Finally, we simplify the equation. y' = 5cos(x) + 2sin(x).

Algebra

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Question:

Two trains start travelling north at the same time. Train A is travelling at 20 mi/hr; Train B is travelling at 15 mi/hr. Assuming Train B begins 100 miles farther north than Train A, how long does it take for Train A to catch up to Train B?

Andrew B.

Answer:

The two trains will meet x miles away from Train B's starting point. When they meet, Train B has traveled x miles, and Train A has traveled x+100 miles. The trains will meet y hours after they start. As a reminder, speed = distance / time. Thus, for Train A: 20 = (x + 100) / y , and for Train B: 15 = x / y. Next, we solve for one variable in terms of the other. The simplest way to do this is with our second equation, 15 = x / y. Multiply the equation by y, and we have 15y = x. Now, we replace x in the first equation with our new solution. We now have: 20 = (15y + 100) / y . We will start by multiplying the equation by y. 20y = 15y + 100 Now, we move all instances of the variable y to one side by subtracting 15y from the equation. 5y = 100. Finally, we divide the equation by 5 to solve for y alone. y = 20. We have solved for y! Now, let us go back to the original question. We want to know how long it will take for Train A to catch up to Train B. In our equations, y represents number of hours after the trains start. Therefore, we have the correct number, and need only specify the units of time. Answer: It will take Train A 20 hours to catch up to Train B (or simply, "20 hours").

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