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 #1
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0

Let's denote lengths:

 

AP=x (since it's a multiple of PB)

 

PB=x/3

 

CP=y (since information about this segment isn't given, we denote it with a variable)

 

PD=y/3 (similar logic as for PB)

 

AB=a (what we're solving for)

 

CD=c (not directly needed, but can be helpful for visualization)

 

Apply the Power of a Point Theorem:

 

The Power of a Point Theorem states that for any point P inside a circle, the product of the lengths of the two segments created by drawing secants from that point to the circle is equal. In our case, point P is inside the circle (since chords intersect within the circle), and we can apply the theorem to both secants AB and CD.

 

For secant AB:

 

AP⋅PB=x⋅3x​=3x2​

 

For secant CD:

 

CP⋅PD=y⋅3y​=3y2​

 

Since both expressions represent the same power of point P, they must be equal:

 

3x2​=3y2​

 

Utilize the given information:

 

We are given that AP=3⋅PB, which translates to x=3⋅3x​ (substituting the values we defined). This simplifies to x=3x​, which implies x=0. However, a chord cannot have zero length. Therefore, our initial assumption (that x represents a positive length) must be incorrect.

 

Here's the correction: We can rewrite the given information as x=3PB=3⋅3x​. Solving for x, we get x2=9. Taking the square root of both sides (remembering positive for lengths), we have x=3.

 

Substitute and solve for AB:

 

Since we found x=3, we can substitute this value back into the equation we obtained from the Power of a Point Theorem:

 

3x2​=3y2​

 

332​=3y2​ (substitute x with 3)

 

3=y2

 

Taking the square root (positive for lengths), we have y=3​.

 

Now, consider segment AP : its total length is x+PB=3+33​=4. Since AP=x=3, segment PB must have a length of PB=14−3​=1.

 

Finally, to find the length of AB, we add the lengths AP and PB:

 

AB=AP+PB=3+1=4​.

Mar 20, 2024
 #1
avatar+179 
+1

Analyzing the Cube Cuts

 

Imagine the large cube is 3 units on each side. When we cut it into smaller cubes, each side of the larger cube will be made of 3 smaller cubes.

 

(a) Cubes with One Black Face

 

A cube will have exactly one black face only if it's on the edge of the larger cube but not on a corner.

 

Edges: There are 12 edges on a cube.

 

Corner exceptions: On each edge, there are 2 smaller cubes that touch a corner.

 

Since corner cubes will have 3 black faces, we subtract these exceptions from the total edge cubes. There are 8 corners, so there are 8×2=16 corner exception cubes.

 

Therefore, the number of cubes with one black face is the total number of edge cubes minus the corner exceptions: 12 edges - 16 corner exceptions = -4 cubes

 

This seems like a negative number of cubes, which doesn't make sense. The mistake lies in assuming all the edge cubes have one black face.

 

Here's the correction: We only counted the  edges once, but each edge actually has 2 cubes that qualify (one on each side). So, we need to multiply the number of edges by 2:

 

Total one-black-face cubes = (2 cubes/edge) x (12 edges) - 16 corner exceptions = 24 - 16 = 8 cubes

 

(b) Cubes with No Black Faces

 

These cubes must be completely inside the larger cube, not touching any of the faces.

 

Inner core: Since each side of the larger cube is made of 3 smaller cubes, the inner core will be a cube with sides of length 1 unit less (3 - 2 = 1 unit). The volume of this inner core is therefore 1 x 1 x 1 = 1 cube.

 

Inner cubes: This inner core cube is itself made of smaller cubes. Each side has 1 cube, so there are a total of 1 x 1 x 1 = 1 smaller cube inside.

Therefore, there is only 1 cube with no black faces.

 

(c) Probability of Top Face Black

 

When a small cube is rolled, there are 6 possible faces that could land on top.

 

Out of these 6 faces, only 1 face is painted black (since we're considering the cubes with at least one black face).

 

Therefore, the probability of the top face being black is the number of black faces divided by the total number of faces: Probability = 1 black face / 6 total faces = 1/6.

Mar 19, 2024