# Tutor profile: Drew L.

## Questions

### Subject: Basic Math

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

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

### Subject: Calculus

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

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.

### Subject: Algebra

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?

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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