Posts filed under: Q and A

Researchers in Australia experimented to determine if color of a coffee mug affects how people rate the flavor intensity of the coffee. Volunteers were randomly assigned to taste coffee in mugs: some white and some clear. If same type of coffee was used, researchers concluded that rating was significantly higher for those who drank coffee in clear mug. What can be concluded.
A) Color caused the difference and can be generalized to all drinkers
B)Same as A but cannot be generalized to all drinkers

Volunteers are not a random sample, so the results cannot be generalized to all coffee drinkers. There may be something different about people who would volunteer for a coffee drinking study. For example, people who would volunteer for such a study might be more likely to drink a lot of coffee and thus consider themselves able to discern subtle differences in taste.

Think of it this way: the g function is doing SOME AS-YET-UNKNOWN THINGS to (–x + 7) to turn it into (2x + 1). Of the simple mathematical operations probably at play here (addition, subtraction, multiplication, division) what could be going on?

First, the only way you go from –x to 2x is you multiply by –2. So let’s see what happens if we just multiply f(x) by –2.

–2(–x + 7) = 2x – 14

OK, so the first part’s good now, but how can we turn –14 into +1? Well, we don’t want to multiply or divide again because that would screw up the 2x we just nailed down, so why don’t we try adding 15?

2x – 14 + 15 = 2x + 1

Combine the two operations we just did (multiply by –2, add 15) and you have the g function. The function g will multiply its argument by –2, then add 15. Mathematically, we can write that like this:

g(x) = –2x + 15

Now, start from the top and make sure we’re right.

   g(f(x))
= g(–x + 7)             <– substitute (–x + 7) for f(x)
= –2(–x + 7) + 15   <– apply the g function to (–x + 7)
= 2x – 14 + 15
= 2x + 1

It works! Now all we need to do is calculate g(2).

g(2) = –2(2) + 15
g(2) = 11

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I have a question on practice question 7 in “Circles, Radians, and a Little More Trigonometry.”

I solved it a different way, but I’m not sure if I was just lucky to get the correct answer. Basically, I figured that, because one radian is when the arc and radius are the same length, that radians are like proportions. So if arc RQ were equal to 6, it would be 6/6, or one radian. So then I divided π by 6 and concluded that’s how many radians it was.

Does that actually work? Or was I just lucky?

Yes, that 100% works. Nice thinking!

Practice test 8 Calculator #13

First, you can plug in on this one, so if you feel rusty on your exponent rules at all, that’s a good move. Especially on the calculator section. Say, for example, that you plug in 4 for a. Just enter it all into your calculator (you may need to be careful with parentheses in the exponent depending on the kind of calculator you have):

    \begin{align*}4^{-\frac{1}{2}}&=x\\0.5&=x\end{align*}

Now that you know x, plug 0.5 into each answer choice to see which one gives you 4.

A) \sqrt{0.5}\approx 0.707

B) -\sqrt{0.5}\approx -0.707

C) \frac{1}{0.5^2}=4

D) -\frac{1}{0.5^2}=-4

Obviously, C must be the answer.

To solve this algebraically, first start by squaring both sides. Raising a power to a power is the same as multiplying the powers, so that’ll get rid of the 1/2 on the left:

    \begin{align*}\left(a^{-\frac{1}{2}}\right)^2&=x^2\\a^{-1}&=x^2\end{align*}

Now raise both sides to the –1 power to get a truly alone. Remember that a negative exponent is the same as 1 over the positive exponent, so you can transform the right hand side from x^{-2} to \frac{1}{x^2} to finish the problem.

    \begin{align*}\left(a^{-1}\right)^{-1}&=\left(x^2\right)^{-1}\\a&=x^{-2}\\a&=\frac{1}{x^2}\end{align*}

A triangle’s base was increased by 15%. If its area is increased by 38%, what percent was the height of the triangle increased by?

The easiest way to get this question is to plug in! Say the base and height of the original triangle are each 10. The formula for finding the area of a triangle is A=\frac{1}{2}bh, where b and h are the base and height, so the area of our original triangle is A=\frac{1}{2}(10)(10)=50.

Increasing the base by 15% brings it from 10 to 10\times 1.15=11.5. Increasing the area by 38% brings it from 50 to 50\times 1.38=69. Plug those back into the formula to solve for the new height:

    \begin{align*}69&=\frac{1}{2}11.5h\\138&=11.5h\\12&=h\end{align*}

If the original height was 10 and the new height is 12, then the height increased by 20%.

Hi Mike,

Here is a really confusing question from Applerouth’s SAT text:

a = 1.5 x + 1.50
b = 1.25x + 4.50

In the system of equations above, a and b represent the cost, in dollars, of buying x buffalo wings at two different restaurants. What amount of money will get you the same number of buffalo wings at both restaurants?

A) 12
B) 19.5
C) 20
D) 29.5

The answer is A. No idea how to do this.

You have to find this by looking for the number of wings that costs the same at both stores, so set a and b (the costs at each store) equal to each other and then solve for (the number of wings that will make each store’s cost the same).

1.5x + 1.50 = 1.25x + 4.50
0.25x = 3
x = 12

Therefore, buying 12 wings at each store costs the same amount of money. The question appears to ask HOW MUCH money, so to finish the problem you need to plug 12 back in for x in either of the equations. I’ll do the first one:

a = 1.5(12) + 1.50
a = 19.50

So the answer really should be B. The answer would be A if the question asked for the number of wings that cost the same at both stores, but that’s not what the question asks.

One other note: it might make it more clear what’s going on to graph each line. What the graph below shows is that the price (on the y-axis) is cheaper at store a for up to 12 wings, but store b becomes a better deal for 13 or more wings. At the intersection point—12 wings, $19.50—the same number of wings costs the same amount at both stores.

One way to make sure you get questions like these right is to plug in some values to see which equation makes sense. For example, you might choose to plug in 0 for h here because you know that at zero feet above sea level the boiling point should be 212° F.

Choices C and D don’t give you 212 when h = 0, so they’re definitely wrong!

Now plug in 1000 for h. We should expect the right equation to do what the question says—the boiling point should be (212 – 1.84)° F = 210.16° F. Which remaining choice, A or B, does that when you plug in 1000 for h?

Choice A gives you a crazy low number: 212 – 1.84(1000) = –1628.

Choice B does exactly what you want: 212 – (0.00184)(1000) = 210.16

So the answer is B.

To get this without plugging in, you should think about the elements of the language you’re translating into math. You want to start at 212, and subtract 1.84 degrees for every thousand feet (h/1000), so you might write this to start:

image

From there, a little manipulation lands you on the right answer choice:

image

My recommendation, though: plug in. With a little practice you’ll get very fast at it, and then questions like this go from head scratchers to gimmies.

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I’ll draw this as best I can:

image

 

Look OK? Now let me draw a few more segments in blue…

image

See what’s going on there? All of the small triangles in the figure are the same! (You can prove this with triangle similarity/congruence rules easily enough—I won’t spend the time doing so here, though.)

image

We know that the area of the big triangle is (½)(10)(10) = 50. Further, we know that of the 8 small congruent triangles in the figure, 4 of them are inside the square and 4 aren’t. Because those 8 triangles together have an area of 50, the square has an area of half that: 25.

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The thing to remember about functions is that they do the same thing to whatever is inside the parentheses. So don’t worry about the r vs. the h. They could use x, or a little star symbol, or whatever else they want. What matters is that the function f, as defined here, will equal zero when r = 4, or when r = –1.

When we’re told that f(h – 3) = 0, we can conclude that h – 3 must equal either 4 or –1. Therefore, h must equal either 7 or 2.

The graph below might help show what’s going on visually. When the thing in the parentheses next to the f (also known as the argument of the function) equals –1 or 4, the function equals zero. Therefore h – 3 must equal –1 or 4.

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Hi Mike!
A political scientist wants to predict how the residents of New Jersey will react to a new bill. Which study design will provide the most reliable results?
A) Mailing a questionnaire to each of 200 randomly selected residents of NJ
B) Surveying a group of 300 randomly selected NJ residents
C) Interviewing a group of randomly selected students from a NJ public university
D) Surveying 1500 randomly selected US residents

The answer is B, but why wouldn’t it be A?
Thank you!

Let me first answer your question with a question: why wouldn’t it be B? It has all the elements of A that are good (random selection, sufficiently large sample, targeted to the population the researcher cares about) but the sample in B is larger. Why wouldn’t you choose B over A?

I think the real problem with A is that you don’t know how many questionnaires you’re going to get back when you mail 200 out (I wouldn’t be surprised if the return rate on something like that is way low), but I don’t really think a real SAT math question would ask you to think through real-world logistics to that extent.

Luckily, you need not rely only on your intuition about questionnaire return rates—you can also rely on your knowledge that a larger sample size will always provide marginally more reliable results, so even if you don’t see a difference between a mailed questionnaire and a survey, you can choose based on the difference between 200 and 300.

Whenever you have to square both sides to solve, you have to check for extraneous solutions.

image

That tells you m could be 2 or –10, but because part of the solution was squaring both sides, you need to run both possible solutions through the original equation.Try 2 first:

image

That works, now how about –10?

image

Nope. Remember that the square root function √ returns only positive results, so –10 is an extraneous solution that doesn’t work in the original equation. The sum of all solutions is just 2.

One more note here: I often think it’s worth graphing questions like this.

It should be obvious from that graph that there’s only one intersection of the two functions, at x = 2.

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Basically, the question is: how many seconds is h greater than 21? (This tennis ball is being thrown on a planet other than Earth, by the way. I challenge anyone to throw a tennis ball that stays in the air anywhere near as long as this one does.)

To figure it out, solve for the two times the equation equals 21. The first time will be when the ball crosses into view, and the second one will be the time the ball falls out of view. The time in between is the answer you want.

So there you go–the ball is at height 21 at 1 second and at 21 seconds. There are 20 seconds between there, so that’s the amount of time the ball is visible to the kids on the roof.

The other way some folks might choose to solve this is by graphing. If you graph the given function and also a horizontal line at y = 21, you get a nice visual of the ball’s flight, which helps make the 20-second window the ball is visible more intuitive.

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Think of a 5-12-13 triangle (that’s one of the Pythagorean triples you should know).

image

Say angle A measures x°, which would make angle C measure (90 – x)°. (I’m choosing those based on the fact that I already know that the sine of angle C will be 12/13.)

image

Now that we’ve got it set up, all we need to do is SOH-CAH-TOA it: sin x° = 5/13.

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A function will only have that property if it’s a line that passes through the origin. For example, f(x) = 5x has that property:

You can try the same with other linear functions to see why they won’t work. For example, if f(x) = 5x + 2:

Nonlinear functions also won’t work. For example, if f(x) = x^2:

Anyway, now that we know we’re dealing with a linear function through the origin, we can figure out that if f(6) = 12, then the function we’re dealing with must be f(x) = 2x. Therefore, f(2) = 4.

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