CPhill

9)A and B are two points on a unit sphere. We know the space distance between A and B is sqrt2. What is the length of shortest path on the sphere that connects A to B?

By "space distance" I'm assuming "straight-line" distance...if so...

We can find  the angle theta between these points by the Law of Cosines..so we have

2  = 1^2 + 1^2  - 2cos (theta)

0 = - 2cos (theta)

0/-2 =  cos(theta)

arccos (0)  = pi/2 = theta   ...so   theta  =  pi/2

So....the  length, L, of shortest path on the sphere connecting A and B  is

L  =  r (theta)   =  1  (pi/2)   =   pi/2  ≈  1.57

8)Sphere S is tangent to all 12 edges of a cube with edge length 6. Find the volume of the sphere.

The center of the sphere will lie at the center of the cube

Since the sphere is tangent to all edges of the cube, the  distance from the center of the cube to any edge  will be the radius of the sphere and is given by :

r  =   √ [ 3^2  + 3^3]  = √18  

so........the volume of the sphere is

(4/3)pi (√18)^3  =

(4/3)pi * 18 * √18  =

(24pi) * 3√2  =

72√2 pi   units^3  ≈   319.89  units^3

2)A plane cuts through a sphere with diameter 20 cm, but the closest it gets to the center is 3 cm. What is the area of the intersection of the sphere and the plane in sq cm?

Looking at  the cross-section  of the sphere....it has the equation

x^2 + y^2  = 100

The plane is 3 cm from the center....so......the ntersection of this plane and the sphere will form a circle whose squared radius, r^2,  can be  found as :

r^2  = 100 - y^2

r^2  = 100 - (3)^2

r^2 =  100 - 9

r^2  = 91

So....the area  of this intersection  is just     

pi * r^2   = 

pi *91  = 

91 pi  cm^2   ≈  285.88 cm^2

Here's a pic of the cross-section :

10 paintings fall outside the circles....so....50 fall inside...and we have

x(x - 2) + x + 2x + 8  = 50

x^2 -2x + 3x = 42

x^2 + x  - 42  =  0    factor

(x + 7) ( x - 6)   = 0

Setting  both factors to 0 and solving for x gives  x  = -7 (reject) or x  = 6

So...the total number of painitngs from the 20th century  is x * (x - 2) = 6* (6 - 2)  = 24   (1)

And of these,  x = 6  were British    (2)

So....the probability that a painting is British given that it is from the 20th cenury  is  (2) / (1)  =

6  / 24    =    1 / 4    =   25%

Nice, Gavin.....!!!

D: When two separate events, A and B, are independent, P(A|B)=P(A) . This means that the probability that event B occurred first has no effect on the probability of event A occurring next.

The sample space is just the listing of all the possible outcomes....so we have :

{1H, 3H, 5H, 1T, 3T, 5T}

Independent

P(A l B)   = P (A)

We can simplify this as : 

z  =  3 - 2(2)                - 1

      _______    ≈     ______  ≈  0.9121

       √3 - √8              -1.096

So....the floor  of  this    is    0

See the graph here :  https://www.desmos.com/calculator/llpsxm6v5f

The solutions  appear to be

x = 0.5    and     1 ≤ x < 2