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A rectangle in the coordinate plane is shown below.  A line with slope 2 splits this rectangle into two regions with equal area.  What is the y-intercept of this line?  The rectangle has coordinates (-1,-1), (-1,1), (1,1), and (1,-1).    

Those coordinates draw a square, centered on the origin.    

Any straight line through the origin will bisect the square.   

Conversely, any line that bisects the square will pass through the origin.    

Therefore, the y-intercept is  (0, 0)    

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A line and a circle intersect at A and B as shown below. Find the distance between A and B.    
The circle is x^2 + y^2 = 1, and the line is y = x.    

This is an easy one, I don't even have to draw it. 

The circle is centered on the origin with a radius of 1. 

The line crosses through the origin and intersects the circle.  

The distance between A & B is the diameter of the circle.  =  2 units      

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Let P (n) denote the number of factors of n. Find all values of  n which satisfy P(n) = n.    

n =   1     1 factor:    1    

n =   2     2 factors:  1, 2    

n =   3     2 factors    

n =   4     3 factors:  1, 2, 4    

n =   5     2 factors    

n =   6     4 factors:  1, 2, 3, 6    

n =   7     2 factors    

n =   8     4 factors:  1, 2, 4, 8    

n =   9     2 factors    

n = 10     4 factors:  1, 2, 5, 10    

n = 11     2 factors   

n = 12     6 factors:  1, 2, 3, 4, 6, 12    

Those two are all I can think of.  As n gets larger, P(n) sometimes gets larger,     

but I think the trending demonstrates that P(n) is never going to catch up to n.    

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What is sin C in an acute triangle?

\(A=\arcsin \dfrac{2}{5}\\ A=23.5782^{\circ}\\ B=\arcsin \dfrac{2}{\sqrt{7}}\\ B=49.1066^\circ \\ C=180^{\circ }-23.5782^{\circ}-49.1066^\circ\\ \color{red }C=107.3152^{\circ}\\ The\ triangle\ is\ obtuse-angled!\\ \color{blue}\sin C=0.9547\)

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 What is the area of the triangle with side lengths tan A, tan B, and tan C?

\(f_{AC}(x)=\sqrt{9^2-x^2}\\ f_{BC}(x)=\sqrt{6^2-x^2+10x-5^2}\\ 81-x^2=36-x^2+10x-25\\ 10x=81-36+25\\ x_C=\dfrac{81-36+25}{10}\\ x_C=7\\ y_C=\sqrt{9^2-7^2}\\ C\ (7,5.6569)\)

\(\angle B=180^\circ -atan\ \frac{y_c}{x_C-x_B}=180^\circ-atan\ \frac{5.6569}{7-5}\\ \angle B=109.4712^\circ\\ \color{red}tan\ \angle B=-2.8284\\ The\ sides\ of\ a\ real\ triangle\ are\ greater\ than\ zero. \)

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How many positive integers are there whose digits strictly decrease from left to right,     

and have at most one even digit, and the sum of the digits is 6?    

 6 0          This is legit, because zero isn't even.  Somebody might argue definitions.    

 5 1    

 5 1 0    

 4 2           Won't work, problem says there can't be more than one even digit.    

 3 2 1    

 3 2 1 0    

By the terms of the problem, I find 5 of them.    

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The answer is 24.  I hope this helps!

The answer is 3.  I hope this helps!

Find the area of the triangle with side lengths 4, 5, and 8.    

I've always been sort of partial to Heron's formula.    

Area = sqrt[(s)(s–a)(s–b)(s–c)]    where s = (1/2)(a+b+c)    

s = (1/2)(17) = 8.5     therefore A = sqrt(8.5 x 4.5 x 3.5 x 0.5)    

                                                 A = sqrt(66.9375)    

                                                 A = 8.1815    

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The number of ways is 14.

Find the positive difference between the solutions to the equation 6t^2 + 30 = 41t    

                                     6t² + 30  =  41t    

                                     6t² – 41t + 30  =  0    

this will factor              (6t – 5)(t – 6)  =  0    

                                       t – 6 = 0     ——>     t  =  6    

                                     6t – 5 = 0     ——>     t  =  ⁵/₆    

difference between                                             5 ¹/₆    

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Find the area of the region enclosed by the graph of x^2 + y^2 = 2x - 6y + 6 + 4x - 8y.    

                                      x² + y²  =  2x – 6y + 6 + 4x – 8y    

rearrange & group         (x² – 6x    ) + (y² + 14y    )  =  6    

Complete the squares on x and y.  Add the same amount to the right side that you add to the left.    

                                           (x² – 6x + 9) + (y² + 14y + 49)  =  6 + 9 + 49    

                                           (x – 3)² + (y + 7)²  =  64    

That's a circle in the form   (x – h)² + (y –k)² = r²  where h and k locate the center and r is the radius.    

Area of the circle is A = π r²     ——>>  A = 64 π  =  201.06    

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A committee is to be made up from a group of eight candidates. The committee must consist of at least two people, but at most four people. How many ways can the committee be chosen?    

₈C₂  =  28    

₈C₃  =  56    

₈C₄  =  70    

Add up the possibilities.    

The total comes to 154    

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The area of Kaitlyn's square garden is 360 ft2. One side of the garden is next to a shed. She wants to put a fence around the other three sides of the garden. Determine the approximate amount of fence it will take.    

sqrt(360)  =  approx  19     the length of one side of the garden    

      3 x 19  =  57 ft    

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Given that a^2-b^2=704 and a+b=44, find the value of a-b.    

Write down the first eq                   a² – b²  =  704    

Factor it to                             (a + b)(a – b)  =  704     (eq 1)    

Write down the other eq        (a + b)            =    44     (eq 2)    

See where this is going?    

Divide the left   part of eq 2 into the   left part of eq 1    

Divide the right part of eq 2 into the right part of eq 2    

This gives us (a – b) on the left and 16 on the right    

They're equal, so a – b = 16    

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How many sides has an equiangular polygon if the sum of its exterior angles is equal to the sum of its interior angles?​    

The sum of the exterior angles of any regular polygon is 360^o      

Therefore the sum of the interior angles of our polygon is 360^o    

That makes it a square, which has four sides    

I just recognized it when I saw it, but for a difficult problem, we could have used the formula.   

The sum of the interior angles of a polygon with 'n' sides is given by the formula: S = (n – 2) • 180^o.    

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On the graph of y=(x+2)^2-100, how many points are there whose coordinates are both negative integers?    

Let's change "points" to "integers" ... remembering that a line contains infinite points.    

Any x that is –11 < x < –1 will give a negative y.  Looks like there are 11 of them.    

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You are using cones of varying sizes to serve ice cream in the ice cream parlor where you work. One of the cones is 10 centimeters tall and has a diameter of 5 centimeters.Rounded to the nearest tenth of a cubic centimeter, how much ice cream will completely fill the ice cream cone?    

V  =  (1/3)(π r²)(h)    

V  =  (1/3)(3.14159)(2.5²)(10)    

V  =  (1/3)(3.14159)(6.25)(10)    

V  =  65.44979  rounded to the tenth is  65.4    

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1 out of 6 Nummy Cereal boxes has a bobblehead of Ricky Gervais inside. If Katie buys 3 Nummy Cereal boxes, what is the probability that at least one will have a Ricky Gervais bobblehead inside? Express your answer as a common fraction.     

(1 / 6) • 3  =  3 / 6  =  1 / 2    

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You go for a bike ride. You ride your bike straight at 10 miles-per-hour for 90 minutes.You then turn around to travel back to your original starting point. If it takes you 2 hours to get back, what was your average speed on your way back?    

(10 ^mi/_hr) • (1.5 hr)  =  15 mi    

(15 mi) / (2 hr)  =  7.5 ^mi/_hr    

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If the set has only 1 odd number, its sum will be odd — not divisible by 2. But as soon as there are 2 numbers in the set (whether 2 odds or one even), then some subset will have an even sum.

So, the minimum number of elements needed in the set to guarantee a non-empty subset with a sum divisible by 2 is 2.

Answer: 2

Salem is making sugar cookies. He needs 2 4 /6 cups of white sugar and 1 1/ 6 cups of brown sugar. How much total sugar does Salem need to make his sugar cookies?    

                            2 ⁴/₆   cups of white sugar    

                            1 ¹/₆   cups of brown sugar    

added together    3 ⁵/₆   cups total sugar     

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sally spins a fair spinner numbered 1-6 and flips a fair coin. what is the probability of obtaining a multiple of 2 and a tail   

probability of landing on the tail is            1 / 2    

probability of landing on a 2, 4, or 6 is     1 / 2     (it's 3 / 6 but that's the same as 1 / 2)    

probability of both happening     (1 / 2)(1 / 2)  =  1 / 4    

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For senior night, the vollyball team sold 850 tickets for the big game and made 6050 dollars. If the presale tickets were 5 dollars and the tickets at the door were 9 dollars, how many tickets were sold at the door?    

5P  +  9D  =  6050     (eq 1)     

  P  +    D  =    850     multiply both sides of this times 5    

5P  +  5D  =  4250     (eq 2)    

subtract (eq 2) from (eq 1)    

          5P  +  9D  =  6050    

          5P  +  5D  =  4250    

                     4D  =  1800    

                       D  =    450  at the door    

substitute 450 back into either equation ... pick (eq2, it's easier)    

           P  +  450  =  850    

                       P  =  850 – 450    

                       P  =  400  presale    

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How many positive integers are there whose digits strictly decrease from left to right, and the sum of the digits is 6?    

6    

5 1    

4 2    

3 2 1    

I guess that's it for me.  4    

I can't think of any more.    

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Honestly bro, if there’s no limit on the grid, then it’s impossible to count — because even just two lattice points can form tons of different squares depending on size and rotation. So yeah, there would be infinite such squares.

The answer is 2*n! - n.  Hope this helps!

Ayana has 2.35 in nickels and dimes. If she has 33 coins total, find the number of nickels and dimes.    

5N + 10D  =  235         (eq 1)    

  N +     D  =    33         multiply both sides of this by 5    

5N +   5D  =  165         (eq 2)    

subtract (eq 2) from (eq 1)     5N + 10D  =  235    

                                               5N +   5D  =  165    

                                                          5D  =    70    

                                                            D  =    14 dimes    

substitute into original        N + D    =  33    

                                           N + 14  =  33    

                                           N          =  33 – 14    

                                           N          =  19 nickels    

check answer  (5)(19) + (10)(14)  =?  235     

                              95  +  140        =    235 ... luciendo bien    

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I have 2 standard, 6-sided dice; one is brown, and the other is black. I roll them both. What is the probability that the sum of the numbers that I roll is less than 3?    

The only possible sum less than 3 is 2.    

That is, you have to roll a 1 on each die.    

The chance of rolling two 1's is (1 / 6) • (1 / 6)  =  1 / 36    

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How many ways can Amir, Bin, Charles, and Daria stand in a line if Amir and Bin want to stand next to each other?    

Consider Amir & Bin as a unit.  Then, how many ways can you arrange 3 items.    

The answer to that is 3 • 2 = 6    

Now have A & B swap places.  You can arrange this configuration 6 ways as well.    

Total of ways to arrange them, add both configurations:   6 + 6  =  12    

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The teacher asked Jenny's class to solve the equation 5x - 6 = 4x + 9. Jenny copied the coefficient of  on the left-hand side incorrectly and wrote down a different equation. Jenny correctly determined that her equation had no solutions. What was the coefficient of x on the left-hand side of Jenny's equation?    

4 would do it.  That would create 4x – 6 = 4x + 9    

working problem out                                   4x – 6  =  4x + 9    

next, subtract 4x from both sides and get        – 6  =  + 9      which cannot be    

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Two circles have the same center (Circles which have the same center are called concentric.) The larger circle has radius 10 and the smaller circle has radius 5. Determine the area of the ring between these two circles.    

π r² (larger)  –  π r² (smaller)  =  area of ring    

π • 10² – π • 5²  =  π • 100 – π • 25  =  π • 75    

π • 75 equals approx  235.6    

The area of the ring is the area of the larger circle minus the area of the smaller cirlle.    

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What is the maximum value of 4(x + 7)(2 + x) , over all real numbers x?    

4(x + 7)(2 + x) multiplies out to 4x² + 36x + 56    

The +4x² identifies the expression as a parabola opening upward.     

As such, there is no maximum value of x, it goes on forever.    

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Three candidates were running for office. One of them got half of the vote. Another got two fifths of the vote. What fraction did the third candidate get?    

1 / 2  =  5 / 10    

2 / 5  =  4 / 10     add those two fractions   

             9 / 10     is the total of the first two candidates    

So the third candidate got the other 1 / 10 of the votes    

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A bag contains 6 red marbles, 5 blue marbles and 8 green marbles. If three marbles are drawn out of the bag, what is the probability, to the nearest 10th of a percent, that all three marbles drawn will be green?    

The probability of the 1st marble being green is 8 / 19    

If the 1st marble was green, that leaves 7 in the bag    

The probability of the 2nd marble being green is 7 / 18    

If the 2nd marble was green, that leaves 6 in the bag    

The probability of the 3rd marble being green is 6 / 17    

The probability of all three occurring is (8 / 19) x (7 x 18) x (6 / 17)  =  336 / 5814    

336 / 5814  =  0.05779  which is 5.779%  which rounds to 5.8%    

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Real numbers x and y satisfy    

x + xy = 250y    

x - xy = -240y    

Enter all possible values of x, separated by commas.     

First thing to do is just add them together.    

  x + xy  =    250y    

  x – xy  =  –240y    

2x          =     10y          ——>>     x = 5y    

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Three red and two blue balls are placed in a hat. A ball is chosen at random and not replaced. A second ball is chosen at random. What is the probability that both balls chosen will be blue.    

The 1st ball probability is 2 / 5   

If the first ball is blue, there's only one blue ball in the hat, so ...    

The 2nd ball probability is 1 / 4    

Probability of both happening is (2 / 5) x (1 / 4)  =  2 / 20  =  1 / 10    

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