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Drew L.

Computer Science Student at Princeton with 5 years of Tutoring Experience

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Basic Math

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Question:

$$3 + (2 - 8) / 4 + 8 * 2^2 = $$?

Drew L.

Answer:

When solving such problems, there is an order of operations one must consider which is PEMDAS. P for parenthesis, E for exponent, M is for multiplication, D is for division, A is for addition, and S is for subtraction. Following P, the first step would be to do the operation in the parenthesis first. Which means that the problem will simplify to $$3 + (2 - 8) / 4 + 8 * 2^2 = 3 + (-6)/4 + 8*2^2$$. Following E, you would then do the exponent operation which makes $$3 + (-6)/4 + 8*2^2 = 3 + (-6)/4 + 8*4$$ The next step would be to do M and D so you would simplify the multiplication and the division signs to give $$3 + (-6)/4 + 8*4 = 3 - 1.5 + 32$$ The final step would be to do A and D and add and subtract what is left so $$ 3 - 1.5 + 32 = 33.5$$ which is your final answer

Calculus

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Question:

Find the area under the curve $$y = x ^3 + sinx $$ bounded by $$x = 0, x = 1$$.

Drew L.

Answer:

To determine the area under a curve, you must set up an integral in respect to dx with bounds of x = 0 to x = 1. So, you can set up your equation like $$\int_{0}^{1} x^3 + \sin(x) dx$$. As a intermediate step, you can split this into two integrals to more closely examine the parts of the integral which will give you $$\int_{0}^{1} x^3 + \sin(x) dx = \int_{0}^{1} x^3 dx + \int_{0}^{1} sin(x) dx$$. Looking at the first integral, you know that generally, to integrate an integral of the form $$\int x^n dx$$, the general solution would be $$n*x^{n - 1}$$. So in this case, the exponent of x is 3 so substituting that into the generalized formula will yield $$\int_{0}^{1 } x^3 dx =3x^2|_{0}^{1} = 3(1)^2 - 3(0)^2 = 3$$. As for the integral of $$sin(x)$$, there is simply a formula for it which is: $$\int \sin(x)dx = -cos(x)$$. Solving for the integral of $$sin(x)$$ with the bounds 0 to 1 will simply give you $$-cos(x)|_{0}^{1} = -cos(1) - (-cos(0)) = -cos (1) + 0 = -cos(1)$$. You can now substitute these answer into $$ \int_{0}^{1} x^3 dx + \int_{0}^{1} sin(x) dx$$ which then equals $$3 - cos(1)$$, your final answer.

Algebra

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Question:

Sally bought 80 fruits, spending exactly $150 on apples and oranges for an event. Due to a local shortage of apples, apples are worth $2 and orange are worth only $1. How many of each fruit did she purchase?

Drew L.

Answer:

To solve this problem, you must first translate this word problem’s information into numeric equations. Let’s introduce two variables to represent the number of each fruit that Sally bought: a for apples and o for oranges. If she bought 80 pieces of fruit, then the number of apples plus the number of oranges equals 80. Numerically, you could represent this as a + o = 80. The next piece of information you are given is the price of an apple and an orange as well as the total amount sally spent. This will give 2a + o = 150 as apples are worth 2 dollars, oranges are worth only 1 and together, the amount of money spent on apples and the amount of money spent on oranges equals $150. Now that you have your two equations (1. a + o = 80 2. 2a + o = 150) you can go ahead and solve. One way to solve is through substitution. To do this, take one equation and write it explicitly for a or o. For this solution, we will arbitrarily choose the first equation and the variable a. This means the first step is to solve equation 1 for a -- which you can do by subtracting both sides by o. So: a + o = 80 --> a + o - o = 80 - o --> a = 80 - o Now that you have derived this equation, you can substitute this explicit definition for a into the second equation. So given that a = 80 - o, you can solve for equation 2 like so: 2a + o = 150 --> 2(80 - o) + o = 150 --> 150 --> 160 - 2o + o = 150 --> 160 - o = 150 --> 160 - o - 160 = 150 - 160 --> -o = -10 --> o = 10 So Sally bought 10 oranges. How many apples did she buy? This can be solved by substituting the value of o back into the explicit definition of a (i.e a = 80 - o). So: a = 80 - o --> a = 80 - 10 --> a = 70 So Sally bought 70 apples.

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