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Whatever theorems or steps are necessary.

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If f(x) = 2x+4 then the slope is 2 (since it's already in mx + b form).

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What is your question?

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92/100 = .92

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Paul has 7 of those 9 "chances" so the probability he wins is 7/9. Therefore the probability he doesn't win is 1 - 7/9, or 2/9.

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12.48 is the number (just increasing powers of 2), and the hundredths digit is 8.

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The sum of the interior angles in a quadrilateral is 360 degrees. Thus,

70 + 90 + 2x + 3x = 360

160 + 5x = 360

x = 40

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I would say 1 cm because 1 cm of rain fell...

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...and put them in his Klein bottle.

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(should be x->(2-) but the site won't parse 2-) since all f(x) values where x is less than 2 are equal to 1.

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There is a formula named after Euler that says that F + V - E = 2, for any polyhedron. A cube has six faces, eight vertices, and twelve edges, and satisfies, since 6 + 8 - 12 = 2. An interesting fact is that Euler's formula can be applied to higher dimensions, including 4-, 5-, ... dimensional manifolds.

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The counting numbers 1, 3, 5 add up to 9, which is less than 10. However, if we choose any other set of three distinct counting numbers, the sum would be at least 10 (since we know that {1, 3, 5} has the minimal sum). Therefore {1, 3, 5} is the only such set and the largest of the numbers is 5.

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Not enough information...AB can be anything between 4 and 36 centimeters.

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For each question, you can either answer it true, answer false, or leave it blank. This is 3 ways for each of 10 questions, so the total number of ways is .

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Distance = rate*time only works for constant velocity problems. In this problem, we have the acceleration due to gravity.

The formula for determining position given a constant acceleration is , where a = -9.81 m/s^2, is the original velocity, is the original position (this equation is derived using integral calculus). Here, t = 3 s, = 30 m/s, = 160 m. Therefore,

= 205.9 meters

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2(L+W) where L is the length and W is the width

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T = kh

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Using the identity , if we subtract from both sides, we obtain and we're done.

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I don't think I or any tutor will answer all 11+ parts to this question...but I'll get you started and you can finish the rest.

To find the linear equation, we have two ordered pairs (25, 60) and (15, 75) (here, the price is the x-variable, however normal demand curves list the number sold as the x-variable). The slope m is equal to so m = -3/2. To find the y-intercept replace m, as well as an ordered pair to obtain

60 = (-3/2)(25) + b --> 60 = -75/2 + b --> b = 195/2 = 97.5

The demand equation is therefore given by . You should be able to figure out the rest of the problems. Btw, part i) does not require trial and error, since the revenue is given by px, we have a quadratic equation that we can optimize using the vertex of the quadratic, or by the derivative.

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The 2 in the product 2(n+2) distributes over n and 2, giving 2(n) + 2(2) = 2n + 4.

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The first and fourth equations are equivalent, and they represent direct variation because when x increases, y also increases by the same ratio. The second and third equations are equivalent and they represent inverse variation because when x increases, y decreases (and vice versa).

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Note that this is equivalent to finding the number of ways to make 95, 90, 85, ..., 0 cents using quarters, nickels, and dimes since the pennies will be uniquely determined afterwards. Also, we can further reduce this to make 19, 18, 17, ..., 0 cents using 5 cent, 2 cent, and 1 cent pieces (for example, 3 quarters, 1 dime, 2 nickels making 95 cents is equivalent to 3 5-cent pieces, 1 2-cent piece, 2 1-cent pieces making 19 cents). In essence, we have established a bijective function that maps some set of numbers to a set that is a bit easier to count.

Therefore we want to find the number of ways to make 0 cents, 1 cent, ..., 19 cents (using 5-cent, 2-cent, 1-cent pieces), and add them up. Here, I will demonstrate the number of ways to make x cents *without* using a 5-cent piece:

Case 0: We wish to make 0 cents --> 1 way (just use no coins)
Case 1: We wish to make 1 cent --> 1 way 2*0 + 1*1
Case 2: We wish to make 2 cents --> 2 ways 2*1 + 1*0; 2*0 + 1*2
Case 3: We wish to make 3 cents --> 2 ways 2*1 + 1*1; 2*0 + 1*3
Case 4: We wish to make 4 cents --> 3 ways 2*2 + 1*0; 2*1 + 1*2, 2*0 + 1*4
Case 5: We wish to make 5 cents --> 3 ways 2*2 + 1*1, 2*1 + 1*3, 2*0 + 1*5

There is a pattern, which we can prove by a simple induction or modular arithmetic argument, that the number of ways to make k cents increments by 1 every 2 cents. We can generalize and say that the number of ways to make k cents without using a 5-cent piece is where represents the floor value of z. Summing up from k = 0 to k = 19,

+... + = 110.

Now, suppose and we wish to count the number of ways to make k cents using exactly one 5-cent piece. We can note that this is analogous to computing the ways of making 0 to 14 (subtracting 5 from all terms). In other words, we have

+ ... + = 64

Repeat the same with the cases where two 5-cent pieces and three 5-cent pieces are used. Basically, only the bounds are changed to {0, 9}, {0, 4}.

+ ... + = 30

= 9

Adding all these values up, we get 110 + 64 + 30 + 9 = 213 ways. Since our function is bijective, we can say that the number of ways to make 99 cents is also equal to 213.


This solution required the use of a bijective function, which is normally not taught in a standard middle/high school curriculum. Here is a Wikipedia article in case you're interested: http://en.wikipedia.org/wiki/Bijection

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We have 2(L+H) = 34 --> L+H = 17. So L = 17-H, H = 17-L.

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Width = 1 ft, Length = 5 ft. It's possible to do this without the quadratic formula since the only factors of 5 are 5 and 1, which differ by 4. However the surefire way is to use the quadratic formula.

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No calculator required...we have

-3 + 3y = 2y + 5 Adding -2y to both sdies,

-3 + y = 5

y = 8

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Combine like terms on both sides to get

2u - 9 = (7u - u) + (15 - 20) = 6u - 5

Add -2u to both sides to get

-9 = 4u - 5

Add 5 to both sides to get
-4 = 4u --> u = -1.

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If you recall the 8-15-17 Pythagorean triple, you can conclude the base is 15 and the height is 8.

Or, we can turn this into a quadratic by letting the height be h and the base h+7. Then,







(dividing by 2)



h = 8 (taking the positive solution).

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10 gallons of 2% butterfat. If we mixed some 3.5% butterfat milk, then the average amount of butterfat would exceed 2%.

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Cross multiplying, --> x = 7n.

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In fact you're right...an ordered pair lies on the function if (and only if) equality occurs. Here, if we replace x = -2 and y = 3,

-2 + 2(3) = -2

-2 + 6 = -2, not true, so the ordered pair doesn't lie on the line.