CPhill

Rearrange as

4x^2 - 8x + 4y^2 + 14y  = -12     divide through by 4

x^2  - 2x + y^2 + (7/2)y =  -3      complete the square on x, y

x^2 - 2x + 1  +  y^2 + (7/2)y + 49 /16 =  -3 + 1 + 49/16

(x - 1)^2  + (y + 7/4)^2  = 17 / 16

The radius =   sqrt ( 17/16)  =   sqrt (17) / 4

Angle P =  180 - 52  - 79  =   49

Law of Sines

PQ /sin (52°)  = RQ  / sin (49°)

PQ  / sin (52°)  = 5 / sin (49°) 

PQ  =  5 sin (52°) / sin (49° )  ≈  5.22

Area =   (1/2) RQ * PQ sin ( 79°)  =  (1/2) (5) (5.22) sin (79°)  ≈  12.8  yd^2

https://web2.0calc.com/questions/help-geometry_32272

3. An equilateral triangle is rotated about one of its altitudes, sweeping out a cone in space. If the height of the equilateral triangle is 3sqrt3 then find the volume of the cone.

We can imagine (1/2)  of this triangle  being rotated 360°

This will form a  cone  with a height of  3sqrt (3)

The radius = 3sqrt (3) / sqrt (3)  =  3

The volume is    (1/3)pi ( 3)^2 * 3sqrt (3)  = 

9 sqrt (3) pi 

2. A triangle ABC is rotated around side AB sweeping out a solid. If BC = 3, AC = 4  and AB = 5 then find the volume of the solid.

The area of this triangle is  (1/2) (4)(3)  = 6

The height of the triangle is

6 = (1/2) AB * h

12 = 5 * h

h =12/5 

The volume of 1/2 of this solid will be a cone with a radus of 12/5  and a  height of  5/2

So.....the total volume is    2 (1/3) pi ^ r^2 * h   =

(2/3) pi * ( 12/5)^2 (5/2)  =  48pi / 5

1. Volume of the frustum  =   pi * H / 3 * ( r^2 + Rr + R^2)

     Volume of the cylinder  = pi * R^2 * H

And we  know  that

(7/12) pi R^2 * H =   pi * H / 3 * (r^2 +Rr + R^2)       simplify

(7/12) R^2  =  ( r^2 + Rr + R^2) / 3

(21/12) R^2  = r^2 + Rr + R^2

(7/3) R^2 =  r^2 + Rr + (3/3) R^2

(4/3)R^2  = r^2 + Rr              divide through by r^2

(4/3)R^2 / r^2  =  1 + R/r

(4/3) R^2 / r^2 - R/r  - 1  =   0        mulyiply through by 3/4  and  rearrange

R^2/r^2 - (3/4)R/r   =  3/4             let  R/r =  x

x^2  - (3/4)x  =  3/4              complete the square on  x

x^2 - (3/4)x  + 9/64  = 3/4 + 9/64

(x - 3/8)^2  =  57/64        take the positive root

x - (3/8) = sqrt (57) / 8

x =  [ 3 + sqrt (57) ] /  8

R / r  =  [ 3 + sqrt 57 ]  / 8  so......   

r / R   =   8 / [ 3 + sqrt (57) ] =   8 [ 3  -sqrt (57) ] / [ 9 - 57]  =  8 [ sqrt (57) - 3 ] / [ 48 ] = 

[sqrt (57) -3 ] /  6

(a)

9 + 90*2  + 151 * 3  =  642  digits

(b)  Prob of 2   =

No. of 2's

In the ones place

10 in the first hundred

10 in the second hundred

5 in last 50

= 25

In the tens place

10 in the first hundred

10 in the second hundred

10 in the last 50

= 30

In the hundreds place

51  in  the last 51

Total 2's  = [ 25 + 30 + 51  ]  =  106

Prob of a 2  =  106  /  642  =    53  /  321

6 cubes have 1 black face

12 have 2 black faces

8 have 3 black faces

1 has no black faces (in the center)

Probability of  black face  =

(6/27) ( 1/6) + (12/27)(2/6) + ( 8/27) (3/6)  + (1/27) (0/6)   = 1/3

(The first set of parentheses contains the probability of that type of cube is chosen; the second set of parentheses contains the probability of having a black side facing up; mult/add these together for the total probability

Construct a circle centered at the origin with a radius of 1

The equation of this  circle  is

x^2 + y^2   = 1         (1)

Let P, Q lie on this circle

Line  through OP  has the equation

y = 4x

Sub this into (1)  to find the x coordinate of P

x^2 + (4x)^2   =1

17x^2  = 1

x^2 = 1/17

x =1/sqrt 17

y = 4/sqrt 17

P = (1/sqrt 17, 4/sqrt 17)

Similarly, line through OQ  has the equation

y =5x

Sub this into (1) to find the x coordinate of Q

x^2 + (5x)^2  =1

26x^2  = 1

x^2 = 1/26

x =1/sqrt 26

y = 5/sqrt 26

Q = ( 1/sqrt 26,  5/sqrt 26)

Slope of  PQ =   

[ 4/sqrt 17 - 5/sqrt 26 ] / [ 1/sqrt 17 - 1/sqrt 26 ]   =   [ 5 sqrt 26 - 20 ] / [sqrt 26 - 4 ]  =

[ 4sqrt 26 - 5sqrt 17 ] / [ sqrt 26 - sqrt 17 ]  =

[ 4sqrt 26 - 5sqrt 17 ] [ sqrt 26 + sqrt 17 ] / 9  =

[4 * 26  - sqrt (442) - 85 ]  / 9  =

[19 - sqrt 442 ]  /  9   ≈  -0.225

https://web2.0calc.com/questions/coordinates_80366