# Tutor profile: Victoria P.

## Questions

### Subject: Folklore and Mythology

For each of the following gods from Greek mythology, list their Roman equivalents: a) Zeus b) Hera c) Apollo d) Hades e) Demeter f) Artemis

a) Jupiter b) Juno c) Apollo - Check with your mythology teacher on this one - some classes refer to the Roman equivalent as "Phoebus Apollo" rather than just "Apollo", but "phoebus" is a Roman epithet meaning bright, much how pop culture often uses "Mighty Zeus" when referring to a character playing Zeus. d) Pluto - Again, check with your mythology teacher to see which names you're responsible for. Some texts refer to him as "Dis Pater", and in later mythology "Plutus" (though this is far less common in my experience). e) Ceres - This is one that many students forget. A trick to remember this is that Demeter/Ceres is the god of the harvest, and "ceres" is the Latin word for grain. Modern English derivatives of "ceres" include cereal, if that is easier.

### Subject: ACT

Given that the function $$f(x)=x^2+7$$ and $$f(2k)=39$$, what is the positive value of $$k$$?

$$\textbf{Background}$$ Being the ACT, this question would be a multiple choice question - as such, you can plug in different choices as a last resort if you check your answer and it is incorrect, or it is not one of the choices on the exam. However, it is not good to rely on this method, so I have formatted the question without choices. $$\textbf{Understanding the Question}$$ This question relies on two key concepts tested on the ACT Math section - function notation and the use of algebra to simplify equations and find a variable. $$\textbf{Function Notation}$$ When function notation ($$f(x)$$) is used, we treat the value, in this case $$2k$$, that replaces $$x$$ to be represented in the following format of equation $$x=2k$$, and then plugged in to the function, in this case $$f(x)=x^2+7$$.This gives us $$f(2k)=(2k)^2+7$$. Because the question told us that $$f(2k)=39$$, we can set the right hand side of both equations equal to each other using the property of transitivity (as both are equal to $$f(2k)$$, they are equal to each other). This gives us $$(2k)^2+7=39$$. $$\textbf{Simplifying the Equation}$$ To find our answer, we must simplify the equation $$(2k)^2+7=39$$ found above using simple algebra to isolate k. First, we can subtract 7 from both sides, giving us $$(2k)^2=32$$. From here, we have two approaches we can take - either squaring $$2k$$, or taking the square root of both sides of the equation. I will show both below. $$\textit{Squaring 2k}$$ Squaring (2k) yields $$4k^2$$ (remember that the exponent of 2 is applied to both the 2 and the k, so your answer would not be $$2k^2$$ - this is a common mistake and will likely be used to get one of the incorrect answer choices on the exam to try and trick you), giving us $$4k^2=32$$. Next, to isolate $$k^2$$ we divide both sides by 4, yielding $$k^2=8$$. Finally, we take the square root of both sides to get $$k=2\sqrt{2} || k=-2\sqrt{2}$$ (4 and 2 are factors of 8, and the square root of 4 is either 2 or -2). As the question asked for the positive result, our answer is $$\bf{k=2\sqrt{2}}$$. $$\textit{Square root of both sides}$$ Taking the square root of $$(2k)^2$$ gives us $$2k$$, and taking the square root of $$32$$ gives us $$4\sqrt{2}$$ or $$-4\sqrt{2}$$ (16 and 2 are factors of 32, and the square root of 16 is either 4 or -4). Thus, we have $$2k=4\sqrt{2}$$ or $$2k=-4\sqrt{2}$$ as our answers. In either case, we can isolate k by dividing both sides by 2, yielding $$k=2\sqrt2 || k=-2\sqrt2$$. As the question asked for the positive value of k, our answer is $$\bf{k=2\sqrt{2}}$$. $$\textbf{Final Answer}$$ $$\bf{k=2\sqrt{2}}$$.

### Subject: Algebra

Please solve the following word problem, and check your answer showing all steps: Ms. Smith held a bake sale to fund the school dance. Cookies were sold for $2.00, and brownies were sold for $3.00. At the end of the day, Ms. Smith sold all and made $78.00 and had sold all 30 items.

$$\textbf{Understand the Word Problem}$$ First, we should translate the word problem to algebraic equations. If we have $$c$$ cookies and $$b$$ brownies, we know that $$c+b=30$$ because Ms. Smith sold 30 items total. To make $78, Ms. Smith sold $$c$$ cookies for $2.00 each and $$b$$ brownies for $3.00 each, giving us the second equation $$2c+3b=78$$. To wrap it up, this yields the following system of equations: $(2c+3b=78$)$(c+b=30$) We have two unknown variables ($$c$$ and $$b$$), and two equations, so we can solve the system of equations. There are a few methods to solve this; if you have time on an exam, try to use more than one in order to check your work. Below, I will show the full steps for substitution, and briefly explain the elimination method. $$\textbf{Method 1: Substitution}$$ To use the substitution method, we isolate a variable in one equation, then plug it back into the other equation. For example, we can isolate $$b$$ from the second equation by subtracting $$c$$ from both sides of the equation, giving us $$b=30-c$$. Now, we can substitute $$30-c$$ (the right side of our equation) in for $$b$$ (the left side of our equation) in the first, untouched equation $$2c+3b=78$$. This gives us $$2c+3(30-c)=78$$. Now, we can use order of operations to isolate $$c$$ from this new equation. First, we have two $$c$$ terms, so we should try to combine them. In order to do this, we must first distribute the 3 across the parentheses, yielding $$2c+90-3c=78$$. Next, we simply combine like terms to get $$-c+90=78$$. Subtracting 90 from both sides to isolate the $$-c$$ gives us $$-c=-12$$. Dividing both sides by -1 to isolate $$c$$ gives us $$c=12$$, so we know that Ms. Smith sold $$\textbf{12 cookies}$$. If the question also required the number of brownies sold, we could plug in this value to any of our original equations, the easiest way being to plug in 12 for c in $$b=30-c$$ to give $$b=30-12$$, which simplifies to $$b=18$$. $$\textbf{Method 2: Elimination}$$ To use the elimination method, we use the same two equations, but this time subtract a multiple of one from the other to cancel out one of the variables, and plug back into either initial equation to find the second variable. For example, we could subtract two times the second equation from the first equation to cancel out the $$c$$'s. This would give us $$b=18$$, which would then be plugged into either of the initial equations to find that $$c=12$$ using similar algebra to that used in substitution. $$\textbf{Check Your Work}$$ In order to check your answer (regardless of method used), it helps to plug both values back into the two initial equations. Plugging in 12 for c and 18 for b satisfies both $$2c+3b=78$$ and $$c+b=30$$ because $$2(12)+3(18)=24+54=78$$ and $$12+18=30$$. $$\textbf{Final Answer}$$ Make sure you always pay attention to what the question is asking. This question asked how many cookies Ms. Smith sold, so make sure you specify that $$\textbf{12 cookies}$$ were sold instead of how many brownies were sold or not including units (in this case, "cookies" is the unit).

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