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Not totally sure about this one, geoNewbie, but here's  what I get

We  have  a known (fixed) side that is  15  units in length   { (1,7)  to  (13,16) }   (5,k)

So   one of these must be true

sqrt [ (1-5)^2  +(7-k)^2]  + sqrt [ ( 13-5)^2  + (16-k)^2] > 15

sqrt  [ ( 13 - 5)^2  + ( 16-k)^2 ] + 15  > sqrt [ ( 1-5)^2  + ( 7 - k)^2 ] 

15  + sqrt [ ( 1-5)^2  + ( 7 - k)^2 ]  >  sqrt  [ ( 13 - 5)^2  + ( 16-k)^2 ]

These are a little  messy to solve,  but  using Wolframalpha, I only get a definite  solution for the  first

k < 10    or  k > 10

So   the points   (5,9)   and (5,11)   seem to  be  the answers

Maybe someone else has a  better solution  ????

This is an AMC 2021 A question

we can set up two equations here:

x = 5y - 13

x + y = 140

now, we can use substitution:

5y - 13 + y = 140

6y - 13 = 140

6y = 173

y = 173/6

x = 667/6

hope this helped! please correct me if i am wrong:)

  \(Compute\ \sin^2 5^\circ + \sin^2 10^\circ + \sin^2 15^\circ + \cdots + \sin^2 90^\circ =\, ? \)

Hello Guest!

\(\sin^2 5^\circ + \sin^2 10^\circ + \sin^2 15^\circ + \cdots + \sin^2 90^\circ =\color{blue}9.5 \)

  !

\(\)

I'm not sure but after the synthetic division, wouldn't the equation be x^3 instead of x^2? Because if you look at the original equation, it starts with x^4, not x^3. So after recollecting the numbers, the equation would be

x^3 - 2x^2 + 8x -16 = 0

From here, it would factor out as,

x^2(x-2) +8(x-2) = 0

(x-2) (x^2 +8) = 0

With the zero product property,

x - 2 = 0

x = 2

x^2 +8 = 0

x^2 = -8

After sqaure rooting each side,

x = 2 sqrt 2i, x = -2 sqrt 2i

So, in total, all the roots would be x = 1, 2, 2 sqrt 2i and -2 sqrt 2i

A construction crew is extending a highway sound barrier that is 13 miles long. The crew builds 1/2 of a mile per week. Write an equation in slope-intercept form that represents the length y​ (in miles) of the barrier after x​ weeks.

Hello Guest!

\(N=\dfrac{W}{t}=\dfrac{\frac{1}{2}ml}{wk}\\ \color{blue}W=W_0 +N\cdot t\)

\(W\ [in\ miles]=13\ miles+\dfrac{\frac{1}{2}\ mile}{week}\cdot t\ [in\ week]\)

\(y\ [in\ miles]=13\ miles+\dfrac{\frac{1}{2}\ mile}{week}\cdot x\ [in\ week]\)

  !

x + y =   2sec A

x - y =  2sin A

(2 / (2secA))^2   +   (( 2sin A)  / 2)^2   =

1 / sec^2A     + sin^2 A   =

cos^2 A   + sin^2  A   =

1

Let the nuber of student tickets  = 190 - x

Let the number of general admission tickets =  x

So 

6x + 4.50 (190 -x) =  996  simplify

6x  +  855  -  4.50x  =  996

6x  -4.50x  =  996  - 855

1.50 x   =  141

x = 141 / 1.50   =  94   = general adm tickets sold

the largest possible value of x is just \(0.\overline{98}\). To convert that into a fraction:

\(100x = 98.\overline{98}\\ x=\overline{98}\\ 99x=98\\ x=\boxed{\frac{98}{99}}\)

I'm pretty sure this is an 2021 AMC 10A question, and since using outside sources to ask for help is prohibited, I really ought to report you to the MAA and have you completely disqualified. However, on a second thought, since you're asking this question here, there wouldn't be any point due to the fact that your score would already be so poor there would be no chance of you moving onto the 2021 AOIME.

So... that was kinda sad. :(((

Hi QuestionMachine!

You have quite a few problems, but I think you are more than capable of solving them, so I'll give you hints. :D

1. The sum of the interior angles of a polygon is 180(n-2), where n is the number of sides. 

2. m

Hi QuestionMachine!

You have quite a few problems, but I think you are more than capable of solving them, so I'll give you hints. :D

1. The sum of the interior angles of a polygon is 180(n-2), where n is the number of sides. 

2. m

omg thx so much cphil

This is a pentagon....the sum of the interior angles=  540

So

5x +2  + 10x - 3  + 8x - 19 + 7x -11 + 13x - 31  = 540    simplify

43x  - 62  = 540

43x = 540 + 62

43x =  602

x=  602/43  = 14

Angle S  = 13 (14)  -  31  =  151°

Ask only one question at a time!

1250 + 27.50 W =   1400 + 20W

(27/50 - 20)W = 1400  -1250

7.50W  =  150

W =  150  /  7.50    =  20   (weeks)

Diverges...geometric....the  common  ratio  is  24/16  =  3/2  = 1.5

do you mean What is the measure of the smallest angle? for the last sentence and if so the the answer is 30 degrees

2^x  = 7

4^(x  -1) =    4^x / 4  =  (2^2)^x / 4 = (2^x)^2  /4  = (7)^2/ 4  =  49/4

So the first step is to find what is 1+2+3+4+...+199+200 you can do that by having pairs of 201 so there are 100 pairs and 201x100=20100 and so there are 20100 numbers in the list.

The second step is to find the sum

and the third step is to divide the sum by the umbers. Use these hints!