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Natalie A.
Math teacher for four years
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Applied Mathematics
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Question:

You work at a farm which has a vertical cylindrical oil tank that is 2.5 feet across on the inside. The depth of the oil in the tank is 5 feet. If 1 cubic foot of space holds 7.48 gallons, about how many gallons of oil are left in the tank?

Natalie A.

We're going to need to remember (or develop) some formulas. I honestly don't remember the formula for the volume of a cylinder, but I don't have to. The volume will be the area of the circle multiplied by the height. The area of a circle is $$\pi*r^2$$. Therefore, the volume of oil we have in the tank will be: $$V=\pi*r^2*h$$. We were given the diameter of the circle (2.5 feet across), so we need to cut that in half to get the radius, which is 1.25 feet. We also know the height is 5 feet, since that is how much oil we have in the tank. $$V=\pi*1.25^2*5$$ $$V=24.54$$ cubic feet. (this is a rounded answer). So consider what the original question was asking, and realize this is not our answer! We want to know how many gallons are in the tank. We are given the ratio: $$\frac{1 cubic foot}{7.78 gallons}$$. We can set up two ratios and figure out what we want to know: $$\frac{1 cubic foot}{7.78 gallons}=\frac{24.54 cubic feet}{x gallons}$$ Use the method you prefer to solve for x, I will use cross multiplication: $$1x=24.54*7.78$$ $$x=190.95$$ cubic feet.

Pre-Algebra
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Question:

Frodo's age right now is exactly 3 times his age 3 years hence minus 3 times his age 3 years ago. How old is Frodo right now?

Natalie A.

When tackling problems like these, it's always a good idea to define your variable(s): $$x$$ is Frodos' age right $$\textit{now}$$. $$(x+3)$$ is Frodo's age three years in the future ("3 years hence"). $$(x-3)$$ is Frodo's age three years ago ("3 years ago"). Now we can set up the equation: $$x=3(x+3)-3(x-3)$$. This is the sentence above translated into math. Now we use algebra to solve for $$x$$. Use the distributive property first, then simplify: $$x=3x+9-3x+9$$ $$x=18$$ So Frodo is 18 right now.

Algebra
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Question:

I love literal equations: I plan on taking over the world. I will require honor (H), determination (D), money (M), friends (F), and a weird ingredient from Star Trek: Vulcans (V). Below is an equation that will equal success (S): $$S=\frac{HD}{2M}+(\frac{1}{5}F+V)^2$$ Solve the equation for F, or figure out how many friends I would need in relation to all of these other variables.

Natalie A.

This is good practice working with inverse operations: what does it take to "undo" each piece in order to isolate F? First step, subtract $$\frac{HD}{2M}$$ from both sides: $$S-\frac{HD}{2M}=(\frac{1}{5}F+V)^2$$ Now we need to take the square root of both sides to "undo" the square on the left side: $$\sqrt{S-\frac{HD}{2M}}=\frac{1}{5}F+V$$ Subtract V from both sides: $$\sqrt{S-\frac{HD}{2M}}-V=\frac{1}{5}F$$ And finally, to undo the $\frac{1}{5}$, we need to multiply both sides by 5. $$5(\sqrt{S-\frac{HD}{2M}}-V)=F$$ So now we know how many friends we need in order to take over the world! It's simple enough: take the square root of the difference of S and HD divided by 2M, subtract V, multiply by 5. *cough*that was rough*cough*

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