learnmgcat

The answer is a = 8 and b = 1.

Let's analyze the situation:

George usually arrives at the train station at 5:00 pm to pick up Ava.

Ava arrived early, so she started walking home an hour earlier, which means she started walking at 4:00 pm.

George and Ava met along the route between the train station and their house.

They arrived home at 4:48 pm, which means they drove for 12 minutes after picking up Ava.

Since they drove for 12 minutes after picking up Ava, Ava must have been walking for 4:48 pm - 12 minutes = 4:36 pm. Therefore, Ava had been walking for 4:36 pm - 4:00 pm = 36 minutes before George picked her up.

Let's analyze the situation:

George usually arrives at the train station at 5:00 pm to pick up Ava.

Ava arrived early, so she started walking home an hour earlier, which means she started walking at 4:00 pm.

George and Ava met along the route between the train station and their house.

They arrived home at 4:48 pm, which means they drove for 12 minutes after picking up Ava.

Since they drove for 12 minutes after picking up Ava, Ava must have been walking for 4:48 pm - 12 minutes = 4:36 pm. Therefore, Ava had been walking for 4:36 pm - 4:00 pm = 36 minutes before George picked her up.

Let's analyze the situation:

George usually arrives at the train station at 5:00 pm to pick up Ava.

Ava arrived early, so she started walking home an hour earlier, which means she started walking at 4:00 pm.

George and Ava met along the route between the train station and their house.

They arrived home at 4:48 pm, which means they drove for 12 minutes after picking up Ava.

Since they drove for 12 minutes after picking up Ava, Ava must have been walking for 4:48 pm - 12 minutes = 4:36 pm. Therefore, Ava had been walking for 4:36 pm - 4:00 pm = 36 minutes before George picked her up.

Here is part (b):

The number of ways to choose three subsets A, B, C of S is 2^n * 2^n * 2^n = 2^(3n).

To find the number of ways that A is a subset of B, and B is a subset of C, we can use the following approach:

Choose a subset C of S. There are 2^n ways to do this.

Choose a subset B of C. There are 2^|C| ways to do this, where |C| is the size of C.

Choose a subset A of B. There are 2^|B| ways to do this, where |B| is the size of B.

Therefore, the number of ways that A is a subset of B, and B is a subset of C is:

∑_{C⊆S} 2^|C| * 2^|B|

where the sum is over all subsets C of S.

We can simplify this expression by noting that for any fixed subset C of S, the number of ways to choose subsets B and A of C such that A is a subset of B is 2^(|C|+1). This is because we can choose any subset B of C, and then choose any subset A of B. There are 2^(|C|+1) ways to do this.

Therefore, the number of ways that A is a subset of B, and B is a subset of C is:

∑_{C⊆S} 2^(|C|+1)

We can evaluate this sum using the following trick:

∑_{C⊆S} 2^(|C|+1) = 2 * ∑_{C⊆S} 2^|C|

The sum on the right-hand side is just the number of subsets of S, which is 2^n. Therefore, we have:

∑_{C⊆S} 2^(|C|+1) = 2 * 2^n = 2^(n+1)

Finally, the probability that A is a subset of B, and B is a subset of C is:

2^(n+1) / 2^(3n) = 1 / 2^(2n-1)

Therefore, the probability that A is a subset of B, and B is a subset of C is 1 / 2^(2n-1).

To solve this problem, we need to understand the relationships within the triangle STU. Here are the steps to find the length of \( SX \):

### Step 1: Analyzing the Triangle

Given:

- \( S \), \( T \), and \( U \) are the vertices of the triangle.

- \( M \) is the midpoint of \( ST \).

- \( N \) is a point on \( TU \) such that \( SN \) is the altitude of the triangle.

- \( ST = SU = 13 \), \( TU = 8 \).

- \( UM \) and \( SN \) intersect at \( X \).

### Step 2: Applying the Median and Altitude Properties

Since \( M \) is the midpoint of \( ST \), \( SM = MT = \frac{13}{2} = 6.5 \).

Also, \( SN \) is an altitude, so it is perpendicular to \( TU \).

### Step 3: Use the Property of the Centroid

In any triangle, the centroid (intersection of the medians) divides each median in a 2:1 ratio. Since \( X \) is the intersection of the medians \( SN \) and \( UM \), it is the centroid of triangle \( STU \).

This implies:

\[
SX = \frac{2}{3} \times SN
\]

where \( SN \) is the altitude from \( S \) to \( TU \).

### Step 4: Calculate SN Using the Area of the Triangle

We use the fact that the area of the triangle can be calculated in two ways:

1. Using base \( TU \) and height \( SN \).

2. Using Heron's formula.

#### Heron's Formula:

First, calculate the semi-perimeter \( s \):

\[
s = \frac{ST + SU + TU}{2} = \frac{13 + 13 + 8}{2} = 17
\]

Then, calculate the area \( \Delta \):

\[
\Delta = \sqrt{s(s - ST)(s - SU)(s - TU)} = \sqrt{17(17 - 13)(17 - 13)(17 - 8)} = \sqrt{17 \times 4 \times 4 \times 9} = \sqrt{2448} = 24
\]

#### Area Using Altitude \( SN \):

The area can also be written as:

\[
\Delta = \frac{1}{2} \times TU \times SN = \frac{1}{2} \times 8 \times SN = 4 \times SN
\]

Equating the two expressions for the area:

\[
24 = 4 \times SN \implies SN = 6
\]

### Step 5: Calculate SX

Now that we know \( SN = 6 \), the length of \( SX \) is:

\[
SX = \frac{2}{3} \times 6 = 4
\]

Thus, the length of \( SX \) is \( \boxed{4} \).

Finding the Measure of Minor Arc TU

Understanding the Problem

We have a circle with center O.

Points T and U lie on the circle.

Point P is outside the circle with PT and PU tangent to the circle.

Angle TPO = 33 degrees.

We need to find the measure of minor arc TU.

Solution

Key Concept: The angle formed by two tangents to a circle from an external point is equal to the difference between the intercepted arcs.

Steps:

Find angle TPU:

Since PT and PU are tangents to the circle, angle PTO and angle PUO are right angles (90 degrees each).

In triangle TPU, the sum of angles is 180 degrees.

So, angle TPU = 180 - angle PTO - angle PUO = 180 - 90 - 90 = 0 degrees.

This means that points P, T, and U are collinear.

Find angle TOP:

Since triangle OTP is a right triangle (OT is a radius perpendicular to tangent PT), angle TOP = 90 - angle TPO = 90 - 33 = 57 degrees.

Find the measure of minor arc TU:

Angle TOP is the central angle subtended by minor arc TU.

The measure of a central angle is equal to the measure of its intercepted arc.

Therefore, the measure of minor arc TU is 57 degrees.

So, the measure of minor arc TU is 57 degrees.

Given points \( A = (4, -4) \), \( B = (3, 8) \), and \( C = (-1, 2) \), and a point \( Q \) such that for any point \( P \) in the plane, the following equation holds:

\[
PA^2 + PB^2 + PC^2 = 3PQ^2 + k
\]

We need to find the constant \( k \).

### Step 1: Write down the expression for the sum of squares of distances

Let \( P = (x, y) \) and \( Q = (x_Q, y_Q) \). The squared distances from \( P \) to the points \( A \), \( B \), and \( C \) are:

\[
PA^2 = (x - 4)^2 + (y + 4)^2
\]

\[
PB^2 = (x - 3)^2 + (y - 8)^2
\]

\[
PC^2 = (x + 1)^2 + (y - 2)^2
\]

The sum of these squared distances is:

\[
PA^2 + PB^2 + PC^2 = \left[(x - 4)^2 + (y + 4)^2\right] + \left[(x - 3)^2 + (y - 8)^2\right] + \left[(x + 1)^2 + (y - 2)^2\right]
\]

Expanding these expressions:

\[
PA^2 = (x^2 - 8x + 16) + (y^2 + 8y + 16)
\]

\[
PB^2 = (x^2 - 6x + 9) + (y^2 - 16y + 64)
\]

\[
PC^2 = (x^2 + 2x + 1) + (y^2 - 4y + 4)
\]

Adding them together:

\[
PA^2 + PB^2 + PC^2 = \left[3x^2 + (-8x - 6x + 2x) + (16 + 9 + 1)\right] + \left[3y^2 + (8y - 16y - 4y) + (16 + 64 + 4)\right]
\]

Simplifying:

\[
PA^2 + PB^2 + PC^2 = 3x^2 - 12x + 26 + 3y^2 - 12y + 84
\]

Thus:

\[
PA^2 + PB^2 + PC^2 = 3(x^2 - 4x + \frac{26}{3}) + 3(y^2 - 4y + 28)
\]

### Step 2: Express the right-hand side

The expression for \( 3PQ^2 \) is:

\[
3PQ^2 = 3\left[(x - x_Q)^2 + (y - y_Q)^2\right] = 3\left[(x^2 - 2xx_Q + x_Q^2) + (y^2 - 2yy_Q + y_Q^2)\right]
\]

Expanding:

\[
3PQ^2 = 3x^2 - 6xx_Q + 3x_Q^2 + 3y^2 - 6yy_Q + 3y_Q^2
\]

### Step 3: Set up the equation

We equate \( PA^2 + PB^2 + PC^2 \) with \( 3PQ^2 + k \):

\[
3x^2 - 12x + 110 + 3y^2 - 12y + 84 = 3x^2 - 6xx_Q + 3x_Q^2 + 3y^2 - 6yy_Q + 3y_Q^2 + k
\]

By comparing coefficients, we get:

\[
-12x = -6xx_Q \quad \text{and} \quad -12y = -6yy_Q
\]

So:

\[
x_Q = 2, \quad y_Q = 2
\]

Now, matching the constant terms:

\[
110 + 84 = 3x_Q^2 + 3y_Q^2 + k
\]

\[
194 = 3(2^2) + 3(2^2) + k = 12 + 12 + k = 24 + k
\]

Thus:

\[
k = 194 - 24 = 170
\]

### Final Answer:

The constant \( k \) is \( \boxed{170} \).

Finding |u+2w|

Understanding the Problem

We are given three pieces of information about complex numbers u and w:

|u| = 5

|w| = 3

|u+w| = 6

We need to find the value of |u+2w|.

Solution Approach

We will use the Law of Cosines for complex numbers to solve this problem.

Applying the Law of Cosines

Let θ be the angle between u and w. Then, we have:

|u+w|^2 = |u|^2 + |w|^2 + 2|u||w|cosθ

Substituting the given values:

6^2 = 5^2 + 3^2 + 253*cosθ

36 = 34 + 30cosθ

cosθ = 1/15

Now, consider |u+2w|^2:

|u+2w|^2 = |u|^2 + (2|w|)^2 + 2|u|(2|w|)cosθ

|u+2w|^2 = 5^2 + (23)^2 + 25*(23)(1/15)

|u+2w|^2 = 25 + 36 + 12

|u+2w|^2 = 73

Therefore, |u+2w| = √73.

So, the value of |u+2w| is √73.

To find the probability that all three pieces of a stick, which is 6 units long, are shorter than 6 units after it is broken at two random points, we can use a geometric interpretation.

1. **Setting Up the Problem:**

- Let the stick start at 0 and end at 6 (it has a total length of \( L = 6 \) units).

- Denote the two break points as \( X_1 \) and \( X_2 \), where \( 0 < X_1 < X_2 < 6 \).

- The resulting pieces of the stick will have lengths:

- Piece 1: from 0 to \( X_1 \) (length \( X_1 \))

- Piece 2: from \( X_1 \) to \( X_2 \) (length \( X_2 - X_1 \))

- Piece 3: from \( X_2 \) to 6 (length \( 6 - X_2 \))

2. **Condition for the Pieces:**

- We need each piece to be less than 6 units in length:

- \( X_1 < 6 \)

- \( X_2 - X_1 < 6 \)

- \( 6 - X_2 < 6 \)

- The inequalities simplify to:

- \( X_1 < 6 \) (which will always be true since \( X_1 < X_2 < 6 \))

- \( X_2 < 6 + X_1 \) (or simply \( X_2 < 6 \) due to initial constraints)

- \( X_2 > 0 \) (will also hold since \( X_2 > X_1 > 0 \))

3. **Valid Conditions:**

- The only actual restriction is that \( X_2 < 6 \) (which is always satisfied because \( 0 < X_1 < X_2 < 6 \)).

- The lengths of the pieces will always be less than the length of the original stick (6 units).

4. **Geometric Approach:**

- We represent the breaking points \( (X_1, X_2) \) in the coordinate plane (in the region where \( 0 < X_1 < X_2 < 6 \)).

- The total area of the triangle formed by these restrictions in this coordinate system is represented by:

\[
\text{Area of valid region} = \frac{1}{2} \times 6 \times 6 = 18
\]

- However, the actual region that must be considered is bounded by the triangle formed by \( (0,0), (6,0), (6,6) \). The area of the square is \( 6 \times 6 = 36 \).

5. **Calculating the Probability:**

- The area of the successful outcomes where all pieces are shorter than 6 units is the same as the area where the inequalities hold. As we analyzed, if we consider the constraints, the condition is satisfied for all relative placements of the breaking points.

- Thus the probability \( P \) of obtaining such a division where each piece is shorter than 6 units is simply:
\[
P = \frac{\text{area of valid choices}}{\text{total area}} = \frac{18}{36} = \frac{1}{2}
\]

Thus, the probability that all three resulting pieces are shorter than 6 units is \(\frac{1}{2}\) or 50%.

(x,y) = (3 + 2*sqrt(2), 7 - 2*sqrt(2))

The minimum value is 5/9.

We put the system in matrix form:

[2 1 0] [t] = 3.15

[0 2 1] [b] = 4.85

[1 2 1] [m] = 5.25

Solving, we get b = 1.40, t = 1.20, m = 1.60.  Therefore, one bagel costs 140 cents.