CPhill

The height of ABC =   8

The midpoint of   (2,0) and (8,0)   =   (5,0)

And the line  connnecting   (0,8) and (5,0)   will  divide the area of the orginal triangle in half

The slope  of this line  is    [ 8 - 0 ] / [ 0 - 5 ]  =  8 / -5  = -8/5

The equation of this line is

y = - (8/5)x + 8

The y intercept  = 8

So....the sum of the slope and y intercept  =  -8/5 + 8  =  32 / 5

Here's a graph :

Your approach is the correct one....!!!!

Let's call the radius of the cone, R

If we bisect the vertex angle, we get a 30-60-90 right triangle  

The side across from the 30° angle  = R

The side across from the 60° angle  =  R*sqrt (3) = cone height

So

Vcone   =  (1/3)pi R^2 * height

Vcone =  (1/3)pi* R^2 * R*sqrt (3)    ......so we have

12288pi = (1/3) pi* R^3 * sqrt(3)

12288 =  [sqrt(3)/ 3] R^3        multiply both sides by  3/sqrt(3     

[ 12288* 3 / sqrt(3) ] = R^3           

[ 12288 sqrt(3) *sqrt(3) / sqrt(3) ]  = R^3

[ 12288 sqrt (3) ]  =  R^3                     Note :    { 12288 = 2^12 * 3 }

[ 2^12 * 3 sqrt(3) ] = R^3

[ 2^12 *  3^(3/2) ] = R^3          take the cube root of both sides

2^4 * 3^1/2   =  R

16sqrt(3) = R

So....the height is   R*sqrt(3)  =  16sqrt(3) * sqrt (3)  =  16 * 3  =  48in

"Guest"   =   EP  "Incognito"

Glad to help, ILP.....!!!!

Subtract seven more than twice a number from the square of one-third of the number to get to zero.

Let N be the number.....

"Seven more than twice a number"  ⇒     2N + 7      (1)

"Square of 1/3 of the number" ⇒   (N/3)^2         (2)

So....we are subtracting  (1) from (2)   and this = 0        .....so....

(N / 3)^2  -  [ 2N + 7]  =   0

Glad to save a life when I can.....

No prob....!!!!

OK....this IS a little tough

A  "fraction of 12" can be represented  as  (1/x)*12 

Taking this away from  17   is      17  -  (1/x)*12

So we have

[ 17 - (1/x)*12 ]    -  (1/3)*17    =   6

The first is definitely arithmetic

The common difference between terms   is  4

The sequence can be expressed as

a_n+1 =  a_n + 4       where   a₁ = - 8

I'm not used to finding an graph for these, but here's a possibilty

y = -8 + 4(x - !)   

y = -8 + 4x - 4

y = 4x - 12       

[ Where x is an integer...and  x = 1  gives the 1st term, -8, x = 2 gives the 2nd term, -4.....etc .]

Here's the graph :    https://www.desmos.com/calculator/pewqtw4vmg

The second is not arithmetic....the difference between 2 and 4  is 2......the difference between 8 and 4 is 4.......thus.....the common difference between terms is not constant as required in an arithmetic series

No prob...!!!!