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A semicircle is exactly half a circle, so take the formula for circumference (C = 2πr) and divide by 2. Since r = 2, you end up with (2π(2))/2 = 2π. That covers the curved part.

You know the straight part has a length of 4 because the radius of the semicircle is 2.

4 + 2π is your answer.

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d = 30 + 2(40 – s)

The machine begins the day with $30 inside, so that’s the “30 +” part. Easy enough.

The variable s is defined as how many sodas the machine has in it, but what we really care about is how many sodas are sold. We know the machine begins the day with 40, so 40 – s should give us the number of sodas sold. (When s = 40, no sodas have been sold; when s = 35, 5 sodas have been sold…)

For each soda that’s sold, the machine should have $2 more, so that’s why “2(40 – s)” is in there.

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Trigonometry does the trick here. Below is that line making a 42° angle with the positive x-axis. I’ve also drawn a dotted segment to make myself a neat little right triangle.

Remember that slope is rise over run—how high the line climbs divided by how far it travels right. In this case, the dotted segment labeled a is the rise and the bottom of the triangle labeled b is the run. And luckily for us, the tangent function calculates that a/b ratio! Remember your SOH-CAH-TOA. Tangent = Opposite/Adjacent.

Just use your calculator to evaluate tan 42°. You’ll get 0.90.

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You can make two equations here. First, you know the total number of marbles is 103, so:

The second equation is more complicated, so let’s do it in parts. First, he gives away 15 red marbles, so he should have r – 15 left. He gives away 2/5 of his blue marbles, so he should have b – 2/5b = 3/5b left.

So the ratio of red marbles he has left to blue marbles he has left (which the question tells us is 3/7) should be:

The question asks how many blue marbles he had originally, so let’s substitute and solve for b. First get r by itself in the first equation:

Now substitute that into the second equation and solve:

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This question comes from my own book, so my tips on how to deal with these can be found in the same chapter. The main key to getting it right is making sure you translate the words into math correctly.

Note that although the question tells you that Tariq makes brownies and Penelope makes cookies, in the end it only asks about “treats,” so we can lump cookies and brownies together.

Tariq makes 30 treats per hour and Penelope makes 48 treats per hour. Together, then, they make 78 treats per hour. We know they both worked for the same amount of hours.

The other key to getting this right is keeping track of the units of the numbers you know. In this case, we have treats and hours for units. We know the number of total treats, and we know the rate of treats per hour. We want the number of hours. How do we set up the equation we need to solve? We need to divide the total number of treats, 312, by the number of treats they made per hour, 78.

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Draw this out. Start with the two points you’re given.

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Now remember that the shape is a rectangle, and that you’re told that point B is on the x-axis. The only way that happens is if B is at (5, 0). Point D, by the same logic, must be at (–3, 2).

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Now draw the rectangle and measure the lengths. The long ends have length 8, and the short ends have length 2.

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Therefore, the perimeter is 8 + 2 + 8 + 2 = 20.

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Start with the second equation, which tells you that t = 4.

If t = 4, then you can rewrite the first equation as follows (and solve):

4u – u = 18
3u = 18
u = 6

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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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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:

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

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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:

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

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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.)

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

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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:

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That works, now how about –10?

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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).

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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.)

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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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