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What is an angle bisector?

Thank you!

Six ways the two can stand nest to each other....   the other two spots have two ways to fill with the other two people

6 * 2 = 12 ways

amount of saline in ingredients = saline in final soln

x = 4%     .04x

10% = 2 quarts = .10(2)

Final solution =  .08 (2 + x)

.04x   + .1(2) = .08 ( 2+x)        solve for x quarts

Triangle side rule says any two sides must sum to more than the third side

25  two other legs must equal to 75      49  and 26

looks like 26 is the shortest of the remaining sides    making the short leg less than 26 will break the rule

SInce it is   x = (y-k)^2 + h       Sideways parabola

You are right...I was using the SURFACE AREA of a cylinder.....   Mistake !   D'Oh!

Thanx !   ~EP

I might as well put my 2 cents worth in as well  

Find the vertex of the graph of the equation x - y^2 + 8y = 13 + 6x - 6y.

\(x - y^2 + 8y = 13 + 6x - 6y\\ -y^2+6y + 8y = 13 + 6x -x\\ - y^2+14y = 5x+13\\ y^2-14y = -5x-13\\ y^2-14y+49 = -5x-13+49\\ (y-7)^2 = -5x+36\\ (y-7)^2 = -5(x+7.2)\\ (y-7)^2 = -4*\frac{5}{4}(x+7.2)\\ \)

This is a sideways parabola opening in the negative x direction.

It has a vertex of (-7.2,7)

It has a focal length of  5/4

So the focal point is  

The length of the latus r****m is 5

LaTex

x - y^2 + 8y = 13 + 6x - 6y\\
 -y^2+6y + 8y = 13 + 6x -x\\
- y^2+14y = 5x+13\\
 y^2-14y = -5x-13\\
 y^2-14y+49 = -5x-13+49\\
(y-7)^2 = -5x+36\\
(y-7)^2 = -5(x+7.2)\\
(y-7)^2 = -4*\frac{5}{4}(x+7.2)\\

 e-2 = 0.7182818284590452

e^-2 = 0.1353352832366127

It is just Pythagoras's theorem.

the x value is the horizontal distance from (0,0)

the y value is the vertical distance from (0,0)

these are at right angles to each other.

If you add the squares of these you get the square of the hypotenuse.   

If the hypotenuse is 1 (that is what unit circle means)  you get

x^2+y^2 = 1  

Maybe you can use better words  

Hi Dudeman,

I can see why but not without a fair amount of effort.

Do you have a short cut method?

Good logic guest but you need to be very careful to read the question properly and answer what is asked.

Hi Orange Juice,

"For starters, only numbers that divide over 2 or 5 are terminating decimals."

What about 4 or 8 or any power of 2 or any power of 5 or any power of 10 ..................   

Ohhh ....  do  you mean the denominator must be a power of 2 or a power of 5 or a power of 10 .........

Mmmm not sure what you mean ..............

4T + 3C = 21

3T + 4C = 16

Multiply the fist one by 3 and the second one by 4

12T+9C=63

12T+16C=64

Subtract the first one FROM the second one

7C = 1

C = 1/7

2C=2/7

check

4T+3/7 = 21

4T = 20 +4/7

T= 5 and 1/7

Put into the second equation

3*(5+1/7)+ 4(1/7)

=15+3/7 + 4/7

=16

Works perfectly.

Your question is not complete

Let our 4 numbers be \(a_1, a_2, a_3, a_4\). Since we know that \(a_1 = a_4\), we replace it, getting us:

\(a_1, a_2, a_3, a_1\)

Therefore, we know that the last three numbers form an arithmetic sequence. We can represent each of the other terms in terms of \(a_2\). We know that the common difference is 6, therefore, we can write our 4 numbers as:

\(a_1, a_2, a_2 + 6, a_2 + 12\)

Since the first number is the same as the last number, we can also replace the first number with the last number, which gets us:

\(a_2 + 12, a_2, a_2 + 6, a_2 + 12\)

Let r be the common ratio between our numbers. Then, we know that:

\(r(a_2 + 12) = a_2,\\ ra_2 = a_2 + 6,\\ \)

Solving this systems of equations by subsitution, we find out that \(a_2 = -4, r = -\frac{1}{2}\). Therefore, our 4 numbers are:

\(8, -4, 2, 8\)

In order to find the area of this region, we have to first conver this equation into the standard circle equation. Moving all terms (except the constant term) to the left handside, and simplifying, we get:

\(x^2 - 6x + y^2 + 14y = 6\)

Then, we can complete the square by using the formula (b/2)^2, and add the coresponding value to both sides. Therefore, we obtain:

\((x-3)^2 + (y + 7)^2 = 64\)

Therefore, this is a circle with center (3, -7), and a radius of 8. Therefore, the area of this circle is \(64 \pi\)

Let us simplify h(x) first. Plugging in the value of g(x) directly into h(x), we get:

h(x) = f(2x + 8)

We then can plug in 2x + 8 into f(x), to get:

h(x) = 3(2x + 8) + 4

h(x) = 6x + 28

In order to find the inverse of a function, we have to switch the x and y values first. Hence, we get:

y = 6x + 28

x = 6y + 28

We then have to solve for y. Solving for y, we get:

x - 28 = 6y

(x-28)/6 = y

Therefore, the inverse function of h(x) is \(h^{-1}(x) = \frac{x-28}{6}\)

\(x = - 1/5 (y-7)^2 + 36/5 \\ vertex\ is\ 36/5,\ 7 \)

What do you conclude that from?

  !

(c) For the first card, there are 3 ways to choose the number.  For the second, there 2 choices, and then there is only 1 choice.  So there are 3*2*1 = 6 ways that the numbers can be chosen.  Doing this for the other attributes, we get 6*6*24*6 ways.  But the order of cards doesn't matter in a set, so we divide by 3!: 6*6*24*6/3! = 864.  So there are 864 sets for part (c).

(d) The cards can have the same number, color, shape, or shading.  If all the colors are the same, then there are 6*24*6/3! = 144 sets.  If all the numbers are the same, then there are 144 sets.  We get the same number for shape and shading, so there are 4*144 = 576 sets for part (d).

(e) We need to choose two of the attributes.  There are C(4,2) = 6 ways of choosing two attributes.  For each of these two attributes, there are 3 options.  For the other two attributes, there are 3 ways of assigning the choices, so there are 6*3*3*3*3 = 486 sets for part (e)

(f) First we choose which attributes are the same.  There are C(4,3) = 4 ways of choosing which attributes are the same.  There are then 3*3*4 = 36 ways to assign which is which for each of these three attributes, and there are 4 ways to assign the choices for the fourth attribute, so there are 4*36*4 = 576 sets for part (f).

x + x-13 + 1/3 x = 85

7/3 x = 98

1/3x = 14

x = 42

x-13 = 29

1/3 x = 14

(42,29,14)

I think PIE would be faster, though this is very rigorous without a doubt.

$\frac{\lfloor \frac{36}{4} \rfloor + \lfloor \frac{36}{6} \rfloor - \lfloor \frac{36}{24}\rfloor}{36}$

$\frac{14}{36}$

$\frac{7}{18}$

I think we don't need the radius, EP.  The problem asks for the solution in terms of pi.  I got 18π as shown below.