CPhill

15

50th term of   -2, 4, 8 , 14

This is an arithmetic series with a  common difference  of 6

So....the 50 th term is

-4  + 6 (49)   =

290

14

We have the following  [geometric ] pattern

22,  17.6 , 14.08

The common ratio  is     17.6 / 22   = 4/5

So.....we want to find this

22(4/5)^n  = 10

(4/5)^n  = 10/22

(4/5)^n  = 5/11      take the log of both sides

log (4/5)^n  = log (5/11)

n * log (4/5)  = log (5/11)

n =  log (5/11)  / log (4/5)  ≈  3.5

Thus...we need to round this up to 4

So....the 4th square will have a side length of  < 10  inches

You computed the cross-product correctly, Julius

Also  the  √74  is correct

The  unit   vector   is

[1 / √74  ]  ( 3, 8 , 1)   =  

(3 / √74  ,   8 / √74 , 1 / √74  )

Here's my attempt at D.....

Surface area  of each  donut hole covered by Akshaj  =  4 pi (10)^2  = 400 pi  mm^2

Surface area  of each donut hole  covered by Theo  = 4 pi (8)^2 =  256 pi     mm^2

Surface area of each donut hole covered by Niraek  =  4 pi (6)^2 =  144 pi   mm^2

Ignoring  the  "pi's"   and prime factoring each number we have that

5^2 * 2^4   =  400

2^8  = 256

2^4 * 3^2   =  144

The LCM  of these three numbers is :

2^8 * 5^2 * 3^2   =   57600

So.....when they all finish a donut hole at the same time :

Akshaj  will have covered    57600 / 400   =  144 donut holes

Theo will have covered  57600 / 256  = 225 donut holes

Niraek will have covered  57600 / 144  =  400 donut holes

EDIT to correct an error!!!!

C, What is the radius, in inches, of a right circular cylinder if the lateral surface area is 24 pi square inches and the volume is 24 pi cubic inches?

If  the  volume is  24 pi  in^3, we have that

24 pi =  pi ^ r^2  * h

Solving for  h  we have that

24 / r^2  = h       (1)

And if the lateral surface area is  24 pi   in^2......we have that

24 pi  =   2 pi * r * h

Sub (1)  for  h  and we have

24 pi  =   2 pi * r  * (24/r^2 )

1  = 2 / r

r  = 2   in

9)The circles x^2 + y^2 = 4 and (x-2)^2 + (y-3)^2 = 7 intersect in two points A and B. Find the slope of AB.

Simplify the second equation

x^2 - 4x + 4  + y^2 - 6y + 9  = 7

x^2 -4x +  y^2  - 6y  =  -6

Add both equations

x^2  - 4x + y^2 - 6y  =  - 6

x^2         + y^2         =    4

______________________

-4x   - 6y  =    -10

Solve this for y

4x +  6y  =  10

2x + 3y  = 5

3y  = 5 - 2x

y = [ 5 - 2x ]  / 3       sub this into  x^2 + y^2  = 4  for y

x^2  + ( [ 5 -2x ] / 3 )^2  =  4

x^2  +  [ 4x^2 - 20x + 25 ] / 9  = 4

9x^2  + 4x^2 - 20x + 25  = 36

13x^2 - 20x - 11  =  0

Solving this equation for  x  produces

x  = [10 + 9√3 ]  / 13      or     x  =   [ 10 - 9√3] / 13

And  when x   = [ 10 - 9√3] / 13, then y = [15 + 6√3 ] / 13

And  when x  = [10 + 9√3] / 13, then y  = [15 - 6√3] / 13

So....the points of intersection are

( [ 10 - 9√3 ] / 13, [15 + 6√3 ] / 13 )     and   ( [ 10 + 9√3 ] / 13, [15 - 6√3 ] / 13 ) 

And  the slope between these points   is

[ ( 15 + 6√3) - ( 15 - 6√3)  ]  /  [ (10 - 9√3)  - (10 + 9√3) ]  =

[12√3 ] / [ (-18√3]  =

-2/3

The independent events are :

Two number cubes are rolled at the same time   [the result on both cubes are independent of one another ]

A spinner is spun and a coin is flipped......each outcome is independent of the other

[Hint :  any time "without replacement"  shows up, the events are usually dependent ]

The perimeter of PQRS is  2 ( 8 +  2)  =  2 (10) =  20 units

So....since the perimeter of ABCD  = 60....the scale factor  of ABCD  to PQRS  =   60 : 20  =  3

Therefore.....AB  = 3(PQ)  = 3 (8)  = 24

And BC  = 3 ( QR)  =  3(2)  = 6

B, A 288 degree circular sector with radius 15 is rolled to form a cone. Find the volume of the cone.

The sector radius will from the slant height of the cone.

The  arc length of the sector will form the base circumference of the cone...its length is :

2 pi ( 15) (288/360)   =    24pi   units

So....the radius of the cone is figured as

24 pi  = 2 pi (r)

12  = r

And the height of the cone is

√ [15^2  -12^2 ] = √ [ 225 - 144 ]  = √81  =  9

So....the volume of the cone is

(1/3) pi (radius)^2  * (height)  =

(1/3) pi  * (12)^2  * 9  =

432 pi  units^2

A, An equilateral triangle of side length 6sqrt3 is rotated about an altitude to form a cone. What is the volume of the cone?

The radius of the cone is      6√3 / 2  = 3√3

The height of the cone is   (√3) * 3√3 =   9

So....the volume of the cone  is

(1/3) pi  * radius ^2  * height  =

(1/3) pi  * ( 27) * 9  =

81 pi  units^3