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Given that the polynomial x^2 - 15x + t = 0 has only positive integer roots, find the average of all distinct possible values of t.      

possible              resulting     

values of x          values of t     

   0   15                     0     

   1   14                   14     

   2   13                   26     

   3   12                   36     

   4   11                   44     

   5   10                   50     

   6     9                   54     

   7     8                   56     

                           ^{–––––}     

total of t's              280     

Avg_t  280 / 8  =  35     

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A ship is traveling in a circle around the outside rim of an island surveying a good place to land. If it is 6 miles from a lighthouse at the center of the island and 120° from due north, how far has the ship traveled from the most northern tip of the island to the nearest tenth?     

(120^o / 360^o) • π d     

(1 / 3) • (3.14 • 12 mi)     

3.14 • 4 mi  =  12.56 mi     

round 12.56 to nearest tenth =  12.6 mi     

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How many of the same digits are found in the base 7 and base 8 representations of 609_{10}?     

609₁₀  =  1530₇  =  1141₈     

Looks like there's just one digit in both, namely 1 and there are 4 of them.     

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\(\color{blac}Find\ \angle CAD\ in\ degrees.\\\)

Thanks webbinsight for the useful suggestions to solve this tricky task!

\(\angle ADB =\angle ADE=\alpha\\ \angle ACB=\angle ACE=\beta\)

The angles intersection AD/EC are twice:

\((120°-\alpha )\ and\ (60°+\alpha )\)

The angles intersection AC/BD are twice:

\((30°+\beta)\ and\ (150°-\beta)\\ \angle CAD=60°+\alpha -\beta\)

In the triangle (intersection AC/BD) - A-D the following applies:

\(\alpha =180°-(150°-\beta )-(60°+\alpha -\beta)\\ \color{blue}\beta = 15°\)

The angles intersection BD/CE are twice:

\((180°-30°-2\beta=120°)\ and\ (60°)\)

In the triangle (intersection AD/CE) -  (intersection CE/BD) - D the following applies:

\((120°)+(60°+\alpha )+(\alpha )=180°\\ \color{blue} \alpha =0\ ?\ ?\ ?\)

I ask everyone for help.

 !

A florist sells some red, yellow and orange roses. She sells 60 red and yellow roses, 70 yellow and orange roses and 80 orange and red roses. Find the number of red roses, the number of yellow roses and the number of orange roses that the florist sells.     

R + Y  =  60     (eq 1)     

Y + O  =  70     (eq 2)     

O + R  =  80     (eq 3)     

subtract (2) from (1)      R + Y  =  60     

                                      Y + O  =  70     

                                      R – O  =  –10     (eq 4)     

add (4) and (3)              R – O  =  –10     

                                      O + R  =  80     

                                           2R  =  70     

                                             R  =  35     

substitute R=35 into (3)     O + 35  =  80     

                                                   O  =  45     

substitute O=45 into (2)     Y + 45  =  70     

                                                   Y  =  25     

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Modify one digit in the number 31743 can make it a multiple of 281.  What is the number after change     

31743 / 281  =  112.96+     Truncate the .96 ... all we want right now is the 112.     

112 x 281  =  Ruh Roh, this isn't going to work, so try 111.     

111 x 281  =  31753     Bingo.  Change the 4 to a 5.     

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Two boys and three girls are going to sit around a table with 5 different chairs. If the two boys want to sit next to Jill, in how many possible ways can they be seated?     

The only way both boys can sit next to Jill at the same time is one on each side of her.     

So, consider every configuration that contains B1 - J - B2 as a unit.  There are 2 more     

girls, so there are 3 "entities" call them.  How many ways to arrange 3 entities.     

This would be expressed as the permutation of three things taken three at a time.     

 3P3  =  6     Now have the two boys change sides of Jill and there are 6 more.     

Add them together and there are 12 total possible ways to seat the people.     

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Solve the system of equations 5x+6y=265x+6y=26 and -x+y=-14−x+y=−14 by combining the equations.    

Consider the     –x + y  =  –14 – x + y    

If you would add x – y to both sides, you get 0 = –14    

However, 0 = –14 cannot be,     

at least, not in our universe,     

so the premise is invalid.    

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y = 14  ( x^2 - x/14  - 1/14)       complete the square on  x

y  =  14 ( x^2 - x/14 + 1/784 - 1/14 - 1/784)

y = 14 ( x^2 - x/14 + 1/784  -  57/784)

y= 14 ( x - 1/28)^2  -  57/56

We have the form

y = a ( x - h)^2 + k

The focus  is found  at    (  h ,  k + 1/(4a) )  =  ( 1/28 , -57/56 + 1/56) =     ( 1/28 , -57/56 + 1/56)  =  ( 1/28 , -1)

28  = (1/2)  (  angle ROB  - angle SOT)

56  =  (3 angle SOT  - angle SOT)

56  = 2 angle SOT

28  = angle SOT

3 * angle SOT  = angle ROB =   84

minor arc RS    =   angle  ROS  =  180 - angle SOT - angle ROB =    180 - 28 - 56  =    68°

GOOD JOB !!!!!

https://web2.0calc.com/questions/geometry_80541

x^2 - 2x + y^2 - 4y = 0

x^2 - 2x + 1 + y^2  - 4y + 4  =  5

(x - 1)^2 + ( y - 2)^2  =  5

max  (x + y)  =   1  +  r cos 45 °  +   2  + r sin 45°  =   

1 + sqrt (5)  (1/sqrt 2)  +  2  +  sqrt (5) (1/sqrt 2)

3 +  2 sqrt (5) / sqrt (2)  =

3 +  sqrt (2) * sqrt (5)  =

3  + 2 sqrt (10) /2    

3 + sqrt (10)

Two runners start at vertices A and B simultaneously and run clockwise around the perimeter of square ABCD at the same speed. A drone flies so that it is always at the midpoint of the two runners.  What is the minimum distance between the two runners?  What is the maximum distance between the two runners?    

See this problem at https://web2.0calc.com/questions/geometry_69250    

There, I solved the problem using trending, and Chris solved it using calculus.    

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The answer is 14.  I hope this helps!

The answer is 13/2.  I hope this helps!