CPhill

The slope of the given line = 1/2

A perpendicular line to this has a negative reciprocal slope  =  -2/1  = -2

So....the equation is

(y - 3)  = -2 ( x - 8)

1. Teesha is in the French club. There are 31 students in the club. The French teacher will pick two students at random to guide visiting students from France. What is the probability that Teesha will be picked as a guide? Hint: Teesha must be picked first or second.

Teesa can be  paired with 30 other students...so...she can appear in 30 possible sets   

The total possible sets = C(31, 2)  = 465

So...the probabilty  that she is picked is   30 / 465  = 2/31

2. What is the probability of rolling a sum of 5 on at least one of two rolls of a pair of number cubes?

A. 1/81  B. 25/1296  C. 2/9  D. 17/81

This equals to   

1  - probability of no 5s  on either roll  

There 4 ways to roll a 5 on one roll....and there are 36 possible outcomes...so 32 of them are not a sum of 5

So...the probability is :

1 - (32/36)(32/36)  =

1 - (32/36)^2  =

17 / 81 

ED  =  (2/3) (20)  =  40/3

ΔABF ~ EDF

AB / BF  = ED / DF

20 / BF  = (40/3) / DF

20 / (40/3)  = BF /DF

60/40 = BF /DF

3/2 = BF /DF

So...there are 5 equal parts to BD  and  BF  is 3 of them...so...

BF   = (3/5)BD

And because  FG is parallel to DC in triangle BDC...then...

FG / BF  = CD /BD

FG / (3/5)BD  = 20 / ( BD)

FG  = (3/5)BD * 20  / BD  =  (3/5) * 20   =  12

Welcome aboard, Roses  !!!

(a)

We can transform the original equation to this :

4x^2  + 25y^2   = 100

The center of this ellipse is  at  (0,0)

So..the translated equation is

4x^2 - 16x  + 25y^2 + 150y  =   -c     complete the square on  x and y

4(x^2 -  4x + 4)  + 25 (y^2 + 6y + 9)  =  -c + 16 + 225

4(x - 2)^2  + 25(y +3 )^2  = 241  - c       divide through by 100

(x - 2)^2 / 25 + (y +3 ) / 4  =    [241 - c ] / 100

We want the right side to =  1...so....c  = 141

The center of this ellipse is at  (2 , -3)   so....this is the transaltion  vector

So  a  = 2   b  = -3    and c  = 141

(b)  The center of the translated ellipse is (2, -3)

The minor axis  is parallel to the y axis

The length of this axis  = 4

And the tangent lines tangent to the ellipse and parallel to the x axis  are 2 units above and below the center.....so....their equations are  y = -3 + 2   ⇒  y  = -1  and

y = -3 - 2  ⇒ y = -5

Here's the graph : https://www.desmos.com/calculator/k2p2x4xh7k

$50,000 for each person....so...for two people $100,000  ⇒  "C"

A segment drawn parallel to the base of a triangle splits the sides proportionately

In triangle AXC....since  segment ZY  is parallel to base XC..so...

XZ : ZA  = YC : AY

4 : 8  = YC  : AY

1 : 2  =  YC : AY 

And in triangle ABC.....segment XY  is parallel to base BC...so...

BX :  AX =  YC  : AY

So

BX : AX  = 1 : 2

BX : [  XZ + ZA ]  = 1 : 2

BX /  [ 4 + 8 ]  = 1 : 2

BX / 12  = 1 : 2

So  BX  =  6

tan  =  opp side /  adjacent side  =  20 / 15  =  4 / 3  ≈   1.33  ⇒ "C"

Sphere volume =  (4/3) pi (5)^3   =  500/3  pi

(1/2) the height of the cylinder is given by  sqrt  [ 5^2 - 3^2 ] = sqrt (16)  = 4

So....the total  height  = 8

Cylinder volume  = pi * (3^2) * ( 8)  =  72 pi

Volume outside the cylinder but inside the sphere  =  pi [ 500 / 3 - 72 ]  =

[ 500 - 216 ] / 3  * pi   =  [284/3] pi

So  W  =   284 / 3

P  =  2DB

This is the 6 month premium

We need to divide this by 6 to get the monthkly premium

P  = 2 * 1.02 *  500  /  6  =  1000* 1.02 / 6  =  1020 / 6 = $170  ⇒  "A"