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The equation of a triangle would be : (base x height)/2 

Please specify what you mean by equivalent formula as this the the best and easiest one to use 

There are 4 ways to choose the box for the black ball.

After we have placed the black ball, there are 3! ways to arrange the white balls in the remaining 3 boxes.

So the total number of ways to place the balls is 4 * 3! = 24.

Since f is its own inverse, then f(f(x)) = x for all x in the domain of f.

If x > 3, then f(x) = k(x) and f(k(x)) = k(k(x)) = x.

If x <= 3, then f(x) = x^2 - 6x + 12 and f(x^2 - 6x + 12) = (x^2 - 6x + 12)^2 - 6(x^2 - 6x + 12) + 12 = x^4 - 12x^3 + 36x^2 - 36x + 144 - 6x^2 + 36x - 72 + 12 = x^4 - 18x^2 + 120 = (x^2 - 6x + 10)^2.

Therefore, we need k(k(x)) = x^2 - 6x + 10 for all x in the domain of f. This means that k(x) must be the inverse of x^2 - 6x + 10.

The inverse of x^2 - 6x + 10 is (x - 2)^2 + 5, so k(x) = (x - 2)^2 + 5.

Therefore, the function k(x) is:

k(x) = (x - 2)^2 + 5

for all x in the domain of f.

(x y)/(x + y) = 1
(x z)/(x + z) = 2
(y z)/(y + z) = 4, solve for x, y, z

Use elimination and substitution to get:

x = 8/5,  y = 8/3,  z=  - 8

First, we can treat the scientists as a single unit, so we have 3 units (the scientists, the mathematicians, and the journalist) to arrange around the table. This can be done in (3−1)!=2!=2 ways.

Next, we need to arrange the scientists and the mathematicians within their respective units. The scientists can be arranged in 5! ways, and the mathematicians can be arranged in 2! ways.

So the total number of arrangements is 2×5!×2!=240.

Leading term: -14x^{16}

Leading coefficient: -14

Degree: 16

Constant term: -2

Coefficient of x^2: -8

(a) Let x be the amount of red potion that must be added to 500 mL of blue potion. The total volume of the mixture will be x+500 mL. The total amount of magical syrup in the mixture will be 0.6x+0.3(500)=0.6x+150 mL. We want the mixture to be 40% magical syrup, so we have the equation:

0.6x+150 = 0.4(x+500)

1.2x = 200

x = 166.67 mL

Therefore, 166.67 mL of red potion must be added to 500 mL of blue potion in order to produce a potion that is 40% magical syrup by volume.

(b) Let x be the amount of red potion and y be the amount of blue potion that are combined to produce 100 mL of a potion that is 54% magical syrup by volume. The total volume of the mixture will be x+y mL. The total amount of magical syrup in the mixture will be 0.6x+0.3y mL. We want the mixture to be 54% magical syrup, so we have the equation:

0.6x+0.3y = 0.54(x+y)

0.06x = 0.18y

x = 3y

Since we want the total volume of the mixture to be 100 mL, we have the equation:

x+y = 100

Substituting x=3y into the second equation, we get:

3y+y = 100

4y = 100

y = 25

Since x=3y, we have x=3(25)=75​.

(c) I'll leave you to think about this one.

(a) Let D, E, and F be the midpoints of BC, AC, and AB, respectively. Then the coordinates of D, E, and F are:

D = (-5, 3) E = (3, -1) F = (-1, 0)

The equation of the median AD is:

y - 4 = \frac{3 - 4}{-5 - 5} (x - 5) y - 4 = -\frac{1}{5} (x - 5) 5y - 20 = -x + 5 x + 5y = 25

The equation of the median BE is:

y - (-1) = \frac{0 - (-1)}{3 - 1} (x - 3) y + 1 = \frac{1}{2} (x - 3) 2y + 2 = x - 3 x - 2y = 5

The equation of the median CF is:

y - 0 = \frac{-4 - 4}{1 - (-9)} (x - 1) y = \frac{-8}{10} (x - 1) y = -\frac{4}{5} (x - 1) 5y = -4x + 5 4x - 5y = -5

(b) Let G be the intersection of two of the medians. We want to show that G lies on the third median.

Without loss of generality, let's assume that G is the intersection of AD and BE. Then the coordinates of G satisfy both the equations of AD and BE.

Substituting the equation of AD into the equation of BE, we get:

x - 2y = 5 4x - 5y = -5

Solving for x and y, we get x=0 and y=1. This means that G is the point (0,1), which is on the third median CF.

Therefore, all three medians pass through the same point.

It is best for these problems to use a picture, and the picture should make it clearer how to proceed. With the given information, I was able to create the following diagram below:

Note that G is the center of the circle. Our goal is to find the length of \(\widehat{DE}\), which is essentially asking for the arc length. The typical arc length formula for degrees is \(S = 2 \pi r * \frac{\theta}{360^{\circ}}\) where S is the arc length, r is the radius of the circle, and \(\theta\) is the angle of the central angle of the arc, which is \(\angle DGE\) in this case.

We can modify this formula somewhat because we already know the circumference. Since the given information states that \(C = 2 \pi r = 24\), we can substitute that into the arc length formula, which turns it into \(S = 24 * \frac{\theta}{360^{\circ}}\).

Now, we must find the central angle, which we can use the Inscribed Angle Theorem to find the measure of the central angle. The Inscribed Angle Theorem states that the measure of the central angle is twice that of the corresponding inscribed angle. Applying that theorem to this particular case, \(2m \angle DFE = m \angle DGE\).

\(2 m \angle DFE = m\angle DGE \\ 2 * 60^\circ = m \angle DGE \\ m \angle DGE = 120^\circ\)

Now that we know the measure of the central angle, we can find the arc length finally.

\(S = 24 * \frac{\theta}{360^{\circ}} \\ S = 24 * \frac{120^{\circ}}{360^{\circ}} \\ S = 24 * \frac{1}{3} \\ S = 8 \text{ units}\)

The question asks to represent the answer as a common fraction, so I suppose \(S = \frac{8}{1} \text{ units}\) would be the right answer in this case.

I am quite impressed with Cphill's procedure! Unfortunately, Cphill made a mistake when he subtracted 8 from both sides instead of adding 8 to both sides! If you correct that mistake, you should get the right answer!

Unfortunately, this answer is not posting because the AI moderation is not posting it. I will add some filler text because that makes the AI believe that I am writing a complete answer. You, as a reader, need not read further that this sentence.

If I add more filler text, then it is more likely that the AI language model will actually post my answer. At least, that is what I think? I hope I am right because I am just trying to point out an error that I found in one of the most recent answers.

@IQantCorrectAnser    Indeed, there might be something wrong with the question, but there's nothing wrong with my answer to the problem stated in this thread.  Maybe the problem on the Staar test says something different and the OP copied it incorrectly.  OP means Original Poster.  

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