CPhill

Thanks for that answer, EP........

Most likely, 60% to 75% of the town rides a bike

We need to  find the distance between

(0,6) and (-4,3)  = √[ ( -4)^2 + (6-3)^2 ] = √ [4^2 + 3^2 ]  =√[25]   = 5

Now, ACG....we could find the lengths of the other two sides and use something known as "Heron's Formula" to calculate the area, but it's messy.....here's an easier way....let the first distance we calculated be  the  base  of the triangle

We need to find the equation of the line between the first two points....

The slope is   [ 6 -3] / [ 0 - -4]  =  3/4

The equation is

y  = (3/4)(x - 0) + 6

y = (3/4)x  + 6      ....... now.... we need this form :  Ax + BY + C  = 0....so ....multiply through by  4.....

4y = 3x + 24

3x  - 4y + 24  = 0

Now the distance  between (2, -2)  and  this line will be the altitude of the triangle and  it is given by

l Ax + By + C l  /  √[ A^2 + B^2 ]       where  (x,y)  = (2, -2)   ....so we have

 l 3(2)  - 4(-2) + 24  l   / √ [ (3)^2 + (-4)^2 ]  =

l 6  + 8  + 24  l  / √25   =

38 / 5     units

So....the area  is

(1/2) base * altitude  =

(1/2) (5) * 38/ 5  =

19 units^2 

1:  The square with vertices (-a, -a), (a, -a), (-a, a), (a, a) is cut by the line y = x/2 into congruent quadrilaterals. The perimeter of one of these congruent quadrilaterals divided by a equals what? Express your answer in simplified radical form.

The length of the sides of one of the quadrilaterals will be the distances between these points :  

(-a,  a)  and  (a, a)  = √ [ -a - a)^2  + ( a - a)^2]  = √ [ (-2a)^2 ]  = 2a

( a,a) and (a, a/2)  = √ [ (a - a)^2  + ( a - a/2)^2 ] = √ [ a^2/4] = a/2

( -a, a)   and ( -a, -a/2)  = √ [ (-a  - -a)^2  + ( a -  -a/2)^2 ] = √ (3/2a)^2  =  (3/2)a

( -a, -a/2)  and ( (a, a/2)  = √ [ (a - -a)^2  + (a/2 -  -a/2)^2  ] = √[ 4a^2  + a^2] = (√5 )a

So...the perimeter of the quadrilateral  divided by  a  =   

[ 2a  + a/2  + (3/2a) + (√5)a ] / a   =   [  4a + (√5)a  ]  / a   =    a [ 4 + √5] / a    =  [ 4  + √5 ]  units

Note, ACG...look at the graph here when a  = 4.....verify for yourself that no matter the value of "a", the answer will be the "constant" answer found above :  https://www.desmos.com/calculator/dfrfzvd5mb

2:  Find the equation of the line passing through the points (-3,-16) and (4,5). Enter your answer in "y = mx + b" form.

Well...this one is a litle easier than the first  !!!

Slope  between the points is  [ -16 - 5 ] / [ -3  -4 ] =  -21 /  -7  = 3

So the equation of the line is :

y  = 3(x - 4)  + 5

y = 3x - 12 + 5

y  = 3x - 7

According to Gay-Lussac's Law :

Pressure  / Temp  = Constant

So

10 atm  / 300 K  =  1/30

So

Pressure  / 100K  = 1/30      multiply both sides by  100

Pressure  =  100/30   =  10/3 =  3.3 atm

Here's another way

The area of the triangle  is   (1/2) product of the legs  = (1/2) (6) (8)  = 24 units^2

So....calling AB the base....the altitude, h,  is : 

24  = (1/2) (10) h

24/5  =  h

Call the altitude, CD

So.....

AD  = sqrt [ AC^2  - CD^2] =

sqrt (6^2  - (24/5)^2)   =

sqrt  (36  - 576/25 ]  =

sqrt [ 324/25]  =

18/5

So....DM  = 5 - 18/5  =  7/5

So  CM  = 

sqrt  ( DM^2  + CD^2 ] = 

sqrt [ (7/5)^2  + (24/ 5)^2 ]  = 

sqrt [ 49  + 576] / 5 =

sqrt [625] /5  =

25 / 5   =

5

4  2  2  2  3  3

4 red, 3 blue , 3 violet

4 red, 2 orange, 2 yellow, 2 green

3 blue, 3 violet, 2 orange, 2 yellow

3 blue, 3 violet, 2 orange, 2 green

3 blue, 3 violet, 2 yellow, 2 green

5 combinations   [ if I counted correctly  !!! ] 

Thanks, tertre!!!....here's one more method

If the number is even and the sum of its digits is divisible by 3, it will be divisible by 6

Thus...we  can ignore any digit, or sums of digits in the number that are themselves divisible by 3

So...we can ignore the leading 9, the 87 that follows, the 6, the 54 that follows and the 3

So...we' are left  with  2A

This is only divible by 6, if A  is even and the sum of 2 + A is divisible by 3

Thus....2 + 4 = 6    is the only possibility.....so....A  = 4

Thanks, tertre....here's another approach

Let x  be the number  of  3 point shots, y be the number of 2 point shots and z be the number of  1 point shots

So....we have this system

3x + 2y + z  = 36    (1)

y  = z     (2)

2x  = z  ⇒  x  = z/ 2     (3)

Sub  (2) amd (3)  into  (1)

3(z/2) + 2z + z  = 36      simplify

(3/2)z + 3z  = 36     multiply through by 2

3z + 6z  =  72

9z  = 72      divide both sides by 9

z  = 8  ⇒   he scored 8 one point shots

According to Charles' Law :

Volume  / Temperature  = Constant

Thus

300  / 30   =  10

Therefore

Volume / 60  = 10      multiply both sides by 60

Volume =  600 L