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"In the triangle shown, for A to be the largest angle it must be that m

You found the "optimal" values for m and n- the largest value for m that satisfies the statement and the smallest value for n that satisfies the statement. It is also correct to say that x must be between 0 and 100 (0

Thank you again Chris

Thank you Chris

Here are all the restrictions we can place on  x  (that I can think of!):

∠A is the largest angle, so the side opposite ∠A is the largest side, so...

x + 9  >  3x

9  >  2x

x  <  \(\frac92\)

x + 9  >  x + 4

9  >  4               (Any value of  x  makes this inequality true.)

The sum of any two sides of a triangle must be greater than the third side, so...

(3x) + (x + 4)  >  x + 9

4x + 4  >  x + 9

3x  >  5

x  >  \(\frac53\)

(3x) + (x + 9)  >  x + 4

4x + 9  >  x + 4

3x  >  -5

x  >  -\(\frac53\)

(x + 9) + (x + 4)  >  3x

2x + 13  >  3x

x  <  13

For A to be the largest angle, it must be that:   x < \(\frac92\)  and  9 > 4  and  x > \(\frac53\)  and  x > -\(\frac53\)  and  x < 13

We can simplify that to say...

For A to be the largest angle, it must be that:   x  <  \(\frac92\)   and   x  >  \(\frac53\)

In other words...

For A to be the largest angle, it must be that:   \(\frac53\)  <  x  <  \(\frac92\)

And       n - m   =   \(\frac92\)  -  \(\frac53\)   =   \(\frac{27}{6}\)  -  \(\frac{10}{6}\)   =   \(\frac{17}{6}\)

But it is confusing that it says, "What is the least possible value of n - m," when it seems like there is only one possible value of  n - m .

Thanks!

We have  C(5,2)   = 10 pairs

They are 

1,2    1,3    1,4   1,5

2,3    2,4    2,5

3,4    3,5    

4,5

The pairs in red have sums < their products

So..the probability is   6 / 10    =  3 / 5

We have that

Δ ADB  ~ Δ AEC      and

Δ ADB  ~ Δ PEB

So

ΔAEC  ~   Δ PEB

AE / EC  = PE / EB

AE / [CP + EP ]    = PE / EB

AE  / [ 10 + 20 ]  = 20 / 30

AE / 30  = 20 / 30  ⇒   AE   = 20

The diagonals bisect each other...therefore......the point where the diagonals intersect  must be the mid-point of these two coordinates  =

  ( [ 1 + 11 ] / 2,  [ -5 + 7 ] / 2 )  =  ( 12 / 2, 2 / 2 )  =  (6,  1 )

(6 , -5)  reflected across the line y = x  ....   reverse the coordinates ⇒  (-5, 6) 

Call the side of the square  = x

Notice that  

 Δ ABE  ~  Δ  EGF

 Δ  EGF ~  Δ  FCD

So

 Δ  ABE ~  Δ FCD 

So

AB / BE   = FC / CD

28 / x  = x / 58   cross-multiply

x^2  = 58 * 28

x^2  = 1624  

So...the area of the square is 1624 units^2

Thank you for trying, but the answer was incorrect :(  Here is the solution I got back in case it helps. 

We approach this problem by trying to find solutions to the equation pq-4p-2q=2 . To do this, we can use Simon's Favorite Factoring Trick and add 8 to both sides to get qp-4p-2q+8=10. This can be factored into

  (p-2)(q-4)=10

We now can see that there are solutions only if p-2 divides 10 . Thus,  there are 4 possible values of p between 1 and 10 inclusive (1,3,4,7) . It follows that the probability of picking such a p is 2/5.

Thanks again!

Thank you so much Omi67

Thank you Chris 

The volume  of the mixture filling the original pan is

9 * 13 * 1   =   117 in^3

1. 5 times this volume   =   175.5 in^3

So

175.5  =  10 * 15   *  depth

175.5 = 150 * depth     divide both sides  by 150

175.5 / 150  = depth  = 1.17 in

There are 3 numbers for p and corresponding 3 numbers for q that will satisfy your equation as follows:

p = (2 (q + 1))/(q - 4) and q must not =4
q = (4 p + 2)/(p - 2) and p must not=2

p =3  and    q=14
p =4  and    q=9
p =7   and   q=6
Therefore the probability is:
3/10 =30%

The perimeter  =   2 [ W + L ] .....where  W and  L   are the dimensions of the fencing

So we have

250  = 2 [ W + L ]     divide both sides by 2

125  = W + L        subtract L  from both sides

125 - L   = W

So...the area  A   = 

A  = W * L       substitute  125 - L  for W.....and we require that

W* L  >   1000

( 125 - L) * L  > 1000 

125L - L^2  > 1000

Sub  x for L   and see the  graph here :

https://www.desmos.com/calculator/jzrsiji5qi

The possible  lengths, L,  of the fence  =   8.59 ft < L < 116.41 ft

Thanks

The slant height of the cone is = BC

The volume of the cone is given by

432pi  = pi/3 * r^2 * height

432pi  = pi / 3 * 12^2  * height

432  = 144/3 * height      multiply both sides by 3/144

3 * 432  / 144  = height 

9 cm  = height

Now...the slant height of the cone  = √ [ r^2 +  h^2 ] = √[12^2 + 9^2] =√225  =15 cm = BC

So...the lateral surface area of the cone = the shaded area of the circle  =

pi * radius * slant height  =     pi * 12 * 15  =   180 pi  cm^2

And we can set up the following relationship

Total circumference of the circle / Total area of the circle = 

Total circumference of shaded area / Total area of shaded area

[2*pi * BC] /[ pi * BC^2]  =  [2pi *  BC * ( D /360)]  / [ 180pi ]

Where D is the number of degrees in the arc of the shaded sector

[2 * pi * 15]  /[ pi * 15^2 ] =[ 2 * pi * 15] * ( D / 360)] /[180pi]

1 / 15^2  =   (D / 360) / 180

1 / 15^2  = D / 64800

64800 / 225  = D = 288°

So...the number of degrees in ABC  = 360  - 288   =  72°

i dont know the answer to 1...

2. a,b,c,d

3. a. 

1- When you retire and wish to withdraw $75,000 per year forever, then that means you must have a "Perpetual Annuity".
2 - The amount of your perpetual annuity would depend on the interest rate you are going to get on that annuity. If we assume that it will be 5% compounded monthly, then the amount of your annuity will come out as follows:

3 - You have to convert your interest rate of 5% compounded monthly to an annual effective rate. And that you do as follows: [1 + 0.05/12]^12 =5.11618978817%.

4-Then you would divide your annual payment of $75,000 by the above annual effective rate and you would get: $75,000 / 0.0511618978817 =$1,465,934.67 - which would be the amount of your perpetual annuity.

5 - In order to save that amount of money over a period of 30 years, or 360 months, you would use this financial formula to accomplish that: FV=P{[1 + R]^N - 1/ R}, Where R=Interest rate per period, N=number of periods, P=periodic payment, FV=Future value.

6 - FV=P x [[1 + R]^N - 1 / R]
$1,465,934.67 = P x [[1 + 0.05/12]^(30*12) - 1 / (0.05/12]]
                              = P x [[1.004166667]^360 - 1 / (0.004166667)]]
                               = P x [3.467744314 / 0.004166667]
                               = P x               832.2586354............
PMT =$1,465,934.67 / 832.2586354
PMT =$1,761.39 - This is the monthly payment that you must save for 360 months to meet your future objective.

There are 9 pizzas so assuming that they are the same price. The equation would be x=85.50/9, when you divide for it you get x as 9.50

1. For a triangle XYZ, we use [XYZ] to denote its area.

Let ABCD be a square with side length 1. Points E and F lie on line BC and line CD, respectively, in such a way that angle EAF=45 degrees If [CEF]=1/9, what is the value of [AEF]?

2. A triangle T is separated into two congruent triangles by a straight cut. Which of the following statements are true?

A) T must have two sides with equal length.
B) The cut must be perpendicular to one side of T.
C)  T must have two interior angles with the same measure.
D) The cut must go through the midpoint of one side of T.

Type your answer as a list of letters separated by commas. For instance, if you believe that conditions A, B, and C are enough, then type "A,B,C" into the answer box. If you believe none of the options are correct, enter "none."

3.   Line PS bisects angle QPR . We can ensure that triangle PQS is congruent to triangle PRS by adding just one more condition. Which of the following statements could be that condition?

A) PQ=PR
B) SQ=SR
C) Angle PQS=Angle PRS
D) Angle PSQ=Angle PSR

Hint: The figure is a kite bisected into two parts from P to S
Type your answer as a list of letters separated by commas. For instance, if you believe that any of the conditions A, B, or C would be enough, then type "A,B,C" into the answer box. If you believe none of the options are correct, enter "none."

it seems the actual question itself is having trouble posting can you post it as a response to this

360° : 12 = 30°

360° : 12 = 30°

I think he changed the wording of the question, here's how i interpreted it:

Let C be the set of all real numbers x such that the largest angle of the triangle in the picture is A. We need to find sup(C)-inf(C).

Also I think he responded in his previous post too.