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First, let's focus on the terms INSIDE of the parenthesis. 

Now, combining the like terms, we get the expression \((3y-11)^{4}\)

Now, since the only way to retrieve y^4 is in the first term, the coefficient is just the coefficient of y to the power of 4. 

THus, we have \(3^4 = 81\)

Now, just for confirmation, we have the full expansion here (no reccomended)

\(81y^{4}-1188y^{3}+6534y^{2}-15972y+14641\)

Thus, the answer is 81. 

Thanks! :)

The vertex of this graph is \((-0.6,11.8)\) 

We start off with the expression \((1+i) \over ( 2-i) (3+i) \)

Now, let's expand and simplify the bottom, we get \((1+i) \over ( 7 -i) \)

Now let's start the rationalization process. Multiply the top and bottom by the conjugate of the denominator, and we get

\(\frac{(1+i)}{ (7-i)} \cdot \frac{(7+i)}{(7+i)} \\ =\frac{6 +8i}{50}\)

Dividing top and bottom by 2 and canceling out the 2, we get

\(\frac{1}{25}( 3 + 4i) \) which is our final answer. 

Thanks! :)

The answer is \(0\)

(15 m * 60 m       *    1 ft /.3048 m   *  1 ft / .3048 m  )  /  ( 400 ft^2 / gal ) = 24.22 gal    ~~   25 gallons required 

Distance = rate X time 

  the distance is the same for both speeds....so equate them 

    40 mi/hr * t   =   55 mi/hr * (t - .41667)                  <==== 25 minutes is .41667 hr

      -15t = -22.9167

            t =1.53 hr 

at 40 mi / hr   the distance is then    40 mi/hr * 1.53 hr = 61.11 miles 

Here, this will help. 

This problem is Problem 9 of the 2012 AIME I Problems. 

Here is a link that gives the answer. 

(0 , 7) 

First, let's distribute and expand the left side of the equation. 

Using the distributive property, we find that

\(x^2+xy=x^2+8\)

Now, notice that the x^2 cancels out. 

\(xy = x^2 - x^2+8\\ xy=8\)

So 8 is our answer. 

Thanks! :)

*1700 points

x = ORIGINAL number of students            3/5 are juniors     2/5 are seniors

then add 12 juniors and subtract 1 senior to get : 

(3/5 x + 12 ) / (2/5 x -1)  = 2/1             cross multiply and solve for x = original # of students = 70 

after adding 11 more = 81 students 

Different name but same problem

it comes from the MathCounts 2022 National Sprint. Question #23.

We need to note a couple of important points to find the equation of this parabola. 

We need one random point along the parabola and the vertex. 

Let's choose the point (0, 7) since it's the y intercept, and the vertex of the graph is \((3, 1)\)

Now, since we have the vertex of the graph, we can put this equation into intercept form. 

We have

\( f(x) = m ( x-3)^2 + 1\) where m is a real number. 

Now, we simply subsitute in the point we retrireved, (0, 7), and plug it in the find m. 

We have that

\(7 = m(0-3)^2+1\\ 7 = 9m+1\\ m = 2/3\)

Now that we have m, we have the full equation of the parabola in INTERCEPT form. 

We need to convert it into standard form to find a,b,c. 

The equation is now

\( f(x) = 2/3(x-3)^2 + 1 \)

This expands to the equation

\(2/3 x^2 -4 + 7 \) meaning that we found each value of a,b, and c. 

We have that \(a = 2/3 , b = - 4 , c = 7 \). Multiplying these 3 together, we get \(2/3*(-4)*7 = -28 *2/3 = -56/3\)

So -56/3 is our final answer. 

Thanks! :)

The denominater is C(64,2)=2016. There are 4 ways to choose a unit cube with 2 painted faces and 24 ways to choose a unit cube with only one painted face, which means probability of choosing one unit cube with 2 painted faces and one unit cube with one painted face is $\frac{24\cdot4}{2016}=\frac{1}{18}$

This is the equation of a circle ....look for the right side of the circle ....

x^2 - 4x + 4      + y^2  - 5y   + 6.25 =   10.25 

(x-2)^2   + ( y- 2.5)^2   = 10.25                This is the cirlce....with center at  2,2.5    and radius sqrt (10.25) 

                                                                      the largest 'x' value will be   2 + sqrt (10.25) = 5.2015 units 

Edith runs  14 km in the time it takes Foley to run 9 km 

Edith has 10 km to go to finish 

10/14  =  x/9         shows   x = distance Foley covers while Edith finishes      6 3/7 km 

           so when Edith finishes , Folyey has covered   6 3/7  + 9 =  15 3/7 km and is thus  8 4/7  km  behind Edith 

Given 

J/5 = D     or  5D = J 

  and 

J/ (5+1) = D - 4     Sub in '5D' for 'J'

5D / 6 = D- 4 

shows D = 24  days 

   24 days x 5 workers = 120 Worker Days for the job 

    if you have 6 workers it will take   120/ 6 = 20 Days     (which is FOUR days less...Check !) 

Now ....how many workers to get the job done in  only  FOUR days  ( which is 20 days earlier) 

120 Worker Days / 4 Days =   30 workers needed         or    hire 25 ADDITIOANL workers to add to the original five 

I can only think of two combinations I can think of are

Combinations of \(2, 2, 4, 5, 5, 5\) and combinations of \(1, 4, 4, 5, 5, 5    \)

Now, we can calculate the number of ways or organize each combination and add them together. 

The number of permutations of 2, 2, 4, 5, 5, 5 is \( 6! = 720 \)  

The number of permutations of 1, 4, 4, 5, 5, 5 is \(6! = 720\)

Now, we had these together to get

\(720 + 720 = 1440 .\)

So I think this is the correct answer, but I'm not actually very sure. 

Thanks! :)