Tutor profile: Nick B.
What is the volume of a box whose length is 10 feet, width is 4 feet, and height is 5 feet?
The first thing I ask myself when answering a question is "what is this question asking for?" In this case, I can see that it is asking for the volume of a box. This leads me to my next set of questions, which are "what additional information does this question give me" and "what is the equation for the volume of a box?" What the Question is Asking: Volume of a box Volume Equation for a box: Length x Width x Height Given Information: Length is 10 feet, width is 4 feet, and height is 5 feet. Answer: Plug in your given information into the equation Volume = 10 feet x 4 feet x 5 feet Volume = 200 ft^3
Find the derivative of d/dx ( sin (4x) + 2x^2)
The easiest way to look at this problem is to look at it as two separate derivatives added together, like this: We now solve for both of these derivatives separately. We will first look at d/dx (sin(4x)) When deriving trig functions, it is important to note that the differential rule is applied here. This means that you have to derive the trig function and the inside separately and multiply together, like this: d/dx (sin(4x)) * d/dx (4x) When deriving the trig function of sin(4x), it is best to pretend the (4x) is not even there and just focus on what the derivative of sin is. We know from trig properties that the derivative of sin is cos, so we can solve this portion of the differential. d/dx (sin(4x)) = cos(4x) Now look at the second part of this and see the d/dx (4x) and solve for it. When looking at derivatives for a single power, the derivative is simply the number in front of the X. In this case it is 4. d/dx (4x) = 4 Now plug these two answers into the equation: d/dx(sin(4x)) * d/dx(4x) cos(4x) * 4 4cos(4x) <---- This is the answer to the first portion! Now we have to look at the second portion of this problem and that is the d/dx (2x^2). The best way to look at this problem is by bring the 2 to the front of the equation to make it: 2* d/dx (x^2). Now solve for d/dx (x^2).This uses the power rule which is [x^n]' = [nx^(n-1)]. In simpler terms, we have x^2 in this problem, so we bring the power of 2 in front of the equation and subtract the 2 by 1. This will make it 2x^1, or simply 2x. Plug this back into the equation of 2 * d/dx (x^2). 2 * (2x) =4x Final Step! Plug your answers for each portion into the original equation: d/dx (sin(4x)) + d/dx (2x^2) = 4cos(4x) + 4x (Final Answer! :D)
Solve the quadratic equation for X: x^2 - x - 12 = 0
Any time I try to solve a quadratic equation, I first try to find the factors of the last number. In this case, it is the number 12. My method here is to start at the number 1, and keep going all the way up until I get to the number 12. For example: 1 x 12 = 12 (This works as a factor!) 2 x 6 = 12 (Factor!) 3 x 4 = 12 (Factor!) 4 x 3 = 12 (this combination is already used in the previous attempt) 5 x _= 12 (NOT a factor) 6 x 2 = 12 (already used) 7 x _ = 12 (NOT a factor) 8 x _ = 12 (NOT a factor) 9 x _ = 12 (NOT a factor) 10 x _ = 12 (NOT a factor) 11 x _ = 12 (NOT a factor) 12 x 1 = 12 (already used) So now I have my list of potential factors being (1 x 12), (2 x 6), and (3 x 4). My next step is to look at the middle number in the equation, which is located directly in front of the letter x. It is very important to see here that the number is actually a -1 and not a 1. Any time it is being subtracted, it is actually a negative number. I now look at list of potential factors and see if I add or subtract a set of numbers, will they add up to my middle number of -1? Since the number is negative, we will subtract the higher number from the lower number. Here we test them out: 1 - 12 = -11 (Doesn't work) 2 - 6 = -4 (Getting closer, but doesn't work) 3 - 4 = -1 (We found our pair!) Now we simply plug in our matching pair into the following format: (X + A) (X - B) = 0 In our case, our A = 3 and our B = -4 (x + 3) (x - 4) = 0 We now have to solve the equation to make it actually be true. We can all agree that any time multiplied by 0 will equal 0. That being said, if either (X + A) equals 0 or (X + B) equals 0, then the equation is correct and will equal 0. Because of this, we can solve (X + 3) = 0 and (X - 4) = 0 separately to get our answers! Final Steps: 1. Write out your first separate equation: X + 3 = 0 2. Get X by itself on one side. In this case you will subtract both sides by 3 to take the 3 to the other side of the equation. X + 3 - 3 = 0 - 3 3. Simplify the equation. X = -3 4. We now have X by itself and have our first answer of X = -3! 5. Repeat steps 1 through 4 for the equation of X - 4 = 0 X - 4 = 0 X - 4 + 4 = 0 + 4 X = 4 (This is our second answer!) Final Answer: X = 4, X = -3
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