The table shows the populations of the San Diego and Detroit metropolitan regions. When were the populations of these regions equal? What was that population?

The bicycle is traveling at a speed of 13 m/s.

In one rotation of the wheel of the bicycle, the distance covered by the bicycle = the circumference of the wheel of the bicycle

Now, according to the question,

Number of rotations of the wheel of the bicycle in 1 minute = 260

∴ Number of rotations of the wheel in 1 second = 260 ÷ 60

                                                                               = 13/3

∴ Distance traveled by bicycle due to the rotation of the wheel in 1 minute = 260 × circumference of the wheel of the bicycle

Or, distance traveled by bicycle in 1 second = 13/3 × circumference of the wheel of the bicycle.

                                                                         = 13/3 × 3 m

                                                                         = 13 m

Hence, the bicycle is traveling at a speed of 13 m/s.

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The complete question is -

How fast is the bicycle traveling if the rear wheel is rotating at a rate of 260 revolutions per minute and the circumference of the wheel is 3 meters.

, X Y is equal to 22.4.

To find X Y, we need to use the given information:

1. W X = 25.3
2. Y Z = 22.4
3. W Z = 25.3

First, let's solve for X. Since W X = 25.3 and W Z = 25.3, we can conclude that X and Z are equal. Therefore, X = Z.

Next, let's solve for Y. Since Y Z = 22.4 and Z is equal to X, we can substitute Z with X in the equation. Therefore, Y X = 22.4.

, X Y is equal to 22.4.

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The average rate of change is 3h² + 12h. Option B

Note that functions are defined as expressions or rules showing the relationship between two variables.

From the information given, we have that;

f(x) = 3x² + 4

The interval { 2 , 2 + h)

Now, substitute the value of x as 2, we have;

f(2) = 3(2)²+ 4

expand the bracket, we have;

f(2)= 12 + 4

f(2) = 16

Then, for x = 2 + h, we have;

f(2 + h) = 3(2+h)² + 4

expand the bracket, we have;

f(2 + h) = 3(4 + 4h + h²) + 4

expand

f(2 + h) = 12 + 12h + 3h² + 4

collect like terms

f(2 + h) = 3h² + 12h + 16

Then,

3h² + 12h + 16 - 16

3h² + 12h

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The measure of angle MRN in rhombus MNOP is 90 degrees.

Quadrilateral MNOP is a rhombus, which means it has four sides of equal length. In a rhombus, opposite angles are congruent. To find the measure of angle MRN, we can use this property.

Step 1: Identify the given information. We know that quadrilateral MNOP is a rhombus.

Step 2: Understand the properties of a rhombus. In a rhombus, opposite sides are parallel and opposite angles are congruent.

Step 3: Determine the relationship between angle MRN and other angles in the rhombus. Since angle MRN is an interior angle, it is supplementary to angle NOP (opposite angle in the rhombus).

This means that the sum of angle MRN and angle NOP is equal to 180 degrees.

Step 4: Calculate the measure of angle NOP. Since quadrilateral MNOP is a rhombus, the opposite angles are congruent. Therefore, the measure of angle NOP is also equal to the measure of angle MRN.

Step 5: Use the relationship between angle MRN and angle NOP. We can set up an equation: MRN + NOP = 180 degrees. Since angle NOP is equal to angle MRN, we can rewrite the equation as: MRN + MRN = 180 degrees.

Step 6: Solve the equation. Combine like terms: 2MRN = 180 degrees. Divide both sides of the equation by 2 to isolate MRN: MRN = 90 degrees.

Therefore, the measure of angle MRN in rhombus MNOP is 90 degrees.

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The amount of Insurance on the home as $216,000, but the amounts of coverage for the garage, loss of use, and personal property cannot be determined without additional information.

To calculate the amounts of coverage for the different components, we need to use the given replacement value and coverage percentages.

a. Amount of insurance on the home:

The amount of insurance on the home can be calculated by multiplying the replacement value by the coverage percentage for the home. In this case, the coverage percentage is 80%.

Amount of insurance on the home = Replacement value * Coverage percentage

Amount of insurance on the home = $270,000 * 80% = $216,000

b. Amount of coverage for the garage:

The amount of coverage for the garage can be calculated in a similar manner. We need to use the replacement value of the garage and the coverage percentage for the garage.

Amount of coverage for the garage = Replacement value of the garage * Coverage percentage for the garage

Since the replacement value of the garage is not given, we cannot determine the exact amount of coverage for the garage with the information provided.

c. Amount of coverage for the loss of use:

The amount of coverage for the loss of use is usually a percentage of the insurance on the home. Since the insurance on the home is $216,000, we can calculate the amount of coverage for the loss of use by multiplying this amount by the coverage percentage for loss of use. However, the percentage for loss of use is not given, so we cannot determine the exact amount of coverage for loss of use with the information provided.

d. Amount of coverage for personal property:

The amount of coverage for personal property can be calculated by multiplying the insurance on the home by the coverage percentage for personal property. Since the insurance on the home is $216,000 and the coverage percentage for personal property is not given, we cannot determine the exact amount of coverage for personal property with the information provided.

the amount of insurance on the home as $216,000, but the amounts of coverage for the garage, loss of use, and personal property cannot be determined without additional information.

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To determine the number of ways the ant can move to the bottom left corner of the 4x6 checkerboard in exactly 10 moves, we can approach this problem using combinatorics and counting techniques.

Let's represent the ant's movements as a sequence of "U" (up), "D" (down), "L" (left), and "R" (right) corresponding to the directions the ant can move. Since the ant needs to reach the bottom left corner in exactly 10 moves, the sequence will consist of 10 characters.

Now, let's count the number of valid sequences. To reach the bottom left corner, the ant needs to move down six times and left four times. Therefore, we need to find the number of different arrangements of six "D" and four "L" in the sequence of 10 moves.

This can be calculated using combinations (binomial coefficients). The formula for combinations is:

C(n, k) = n! / (k! * (n - k)!)

In this case, we need to calculate C(10, 4) since we are selecting 4 positions for "L" from a total of 10 positions.

C(10, 4) = 10! / (4! * (10 - 4)!)

        = 10! / (4! * 6!)

        = (10 * 9 * 8 * 7) / (4 * 3 * 2 * 1)

        = 210

Therefore, there are 210 different ways the ant can move to the bottom left corner of the 4x6 checkerboard in exactly 10 moves.

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The given sequence is 2, 6, 12. To find the common ratio in an explicit formula, we need to determine the relationship between each term in the sequence.

To find the common ratio, we divide each term by the previous term.

Starting with the second term, 6, we divide it by the first term, 2.

[tex]6 / 2 = 3[/tex]

So, the common ratio is 3.

To represent the sequence using an explicit formula, we can use the general form of an explicit formula for geometric sequences, which is:

[tex]a_n = a1 * r^(n-1)[/tex]

Here, "an" represents the nth term in the sequence, "a1" represents the first term, "r" represents the common ratio, and "n" represents the position of the term in the sequence.

Given that the first term (a1) is 2, and the common ratio (r) is 3, the explicit formula for the sequence is:

[tex]a_n = 2 * 3^(n-1)[/tex]

This formula can be used to find the value of any term in the sequence.

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All the students in an algebra class took a 100-point test. Five students scored 100, each student scored at least 60, and the mean score was 76. What is the smallest possible number of students in the class Let the number of students in the class be n. The total marks obtained by all the students = 100n.

The total marks obtained by the five students who scored 100 is 100 x 5 = 500.As per the given condition, each student scored at least 60. Therefore, the minimum possible total marks obtained by n students = 60n.Therefore, 500 + 60n is the minimum possible total marks obtained by n students.

The mean score of all students is 76.Therefore, 76 = (500 + 60n)/n Simplifying the above expression, we get: 76n = 500 + 60n16n = 500n = 31.25 Since the number of students must be a whole number, the smallest possible number of students in the class is 32.Therefore, there are at least 32 students in the class.

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The difference between the largest and smallest number of dimes that could be in the bank is 100.

Let's assume the number of nickels in the piggy bank is N, the number of dimes is D, and the number of quarters is Q.

From the given information, we can form two equations based on the number of coins and the total value:

Equation 1: N + D + Q = 100 (total number of coins)

Equation 2: 0.05N + 0.10D + 0.25Q = 8.35 (total value in dollars)

Now, let's determine the range for the number of dimes, D.

To find the smallest number of dimes, we maximize the number of nickels and quarters, which minimizes the number of dimes. Let's assume all remaining coins (100 - D) are nickels:

Equation 1: D + Q = 100 - N

Equation 2: 0.10D + 0.25Q = 8.35 - 0.05N

Since we want to minimize D, let's consider the maximum values for N and Q. Assuming all remaining coins are nickels, we have N = 100 - D - Q.

Plugging in these values, we get:

0.10D + 0.25Q = 8.35 - 0.05(100 - D - Q)

0.10D + 0.25Q = 8.35 - 5 + 0.05D + 0.05Q

0.05D + 0.20Q = 3.35

To simplify, we multiply the equation by 20:

D + 4Q = 67

The largest value for Q would be when D = 0. Therefore, if we assume all remaining coins are quarters, we have:

D = 0

Q = (100 - D) = 100

So, the largest number of quarters is 100, and the largest number of dimes is 0.

To find the largest value for D, we maximize the number of dimes. Assuming all remaining coins are nickels:

N = 100 - D - Q

Plugging this into Equation 2:

0.10D + 0.25Q = 8.35 - 0.05(100 - D - Q)

0.10D + 0.25Q = 8.35 - 5 + 0.05D + 0.05Q

0.05D + 0.20Q = 3.35

Multiplying by 20:

D + 4Q = 67

The smallest value for Q would be when D = 100. Therefore, if we assume all remaining coins are quarters, we have:

D = 100

Q = (100 - D) = 0

So, the smallest number of quarters is 0, and the smallest number of dimes is 100.

The difference between the largest and smallest number of dimes is:

100 (largest) - 0 (smallest) = 100.

Therefore, the difference between the largest and smallest number of dimes that could be in the bank is 100.

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If you know the measures of two sides and the angle between them, you can use the Law of Cosines to find missing parts of any triangle.

The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It is used to solve triangles when the measures of two sides and the included angle are known, or when the measures of all three sides are known.

The formula for the Law of Cosines is:

c² = a² + b² - 2ab cos(C)

where c is the length of the side opposite angle C, and

          a and b are the lengths of the other two sides.

The Law of Cosines is a powerful tool for solving triangles, particularly when the angles are not right angles. It allows us to determine the unknown sides or angles of a triangle based on the information provided

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The measure of -50° in radians is approximately -0.87π or -2.74.

To convert an angle from degrees to radians, we use the conversion factor that 180 degrees is equal to π radians.

In this case, we have -50°. To find its measure in radians, we can multiply -50° by the conversion factor:

-50° * (π/180°)

Simplifying, we get:

-50π/180

Dividing both numerator and denominator by 10, we have:

-5π/18

Rounded to the nearest hundredth, this is approximately -0.87π.

Alternatively, we can calculate the decimal approximation of the measure in radians. Since π is approximately 3.14159, we can substitute this value:

-5(3.14159)/18

This simplifies to:

-0.87267

Rounded to the nearest hundredth, the measure of -50° in radians is approximately -2.74.

In conclusion, the measure of -50° in radians is approximately -0.87π or -2.74.

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Therefore, approximately 95.82% of the variable lies between 10 and 17.

To find the percentage of the variable that lies between 10 and 17, you can multiply the probability by 100. Given that the probability of the variable lying between 10 and 17 is 0.9582, the percentage can be calculated as follows:

Percentage = Probability * 100

Percentage = 0.9582 * 100

Using a calculator, we find:

Percentage ≈ 95.82%

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The correct order of the values is: -6, |-1.5|, -4, |5|.

x = -2 and the values |-1.5|, -4, |5|, |-6|, we need to order them from least to greatest.

Here are the steps to solve the problem:

Substitute the value of x in each term and simplify:

|-1.5| = 1.5

|5| = 5

|-6| = 6

Substitute the value of x=-2 in the equation:

|-2| = 2

-(-2) = 2

Now, we have the following values: 2, 2, 1.5, 4, 5, and 6.

Sort the values from least to greatest: -6, |-1.5|, -4, |5|.

Therefore, the correct order of the values is: -6, |-1.5|, -4, |5|.

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Error rate refers to the frequency or proportion of errors made in a particular context or process. It is commonly used in various fields such as statistics, computer science, and quality control.

To find the error rate for the training subset, we can use the fact that the average error rate is 0.06.

Let's denote the error rate for the training subset as E_train. We can express the average error rate as:

average error rate = (error rate for training subset + error rate for test subset) / 2

0.06 = (E_train + error rate for test subset) / 2

Multiplying both sides of the equation by 2, we get:

0.12 = E_train + error rate for test subset

Since K=1, the error rate for the test subset would be 0.12 - E_train.

Therefore, we can expect the error rate for the training subset to be 0.12 - E_train.

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y varies directly with x, and the constant of variation is -10.

To determine whether y varies directly with x, we need to check if the equation can be written in the form y = kx, where k is the constant of variation.
In the given equation, y = -10x, we can see that y and x are directly proportional, since the equation can be written in the form y = kx.
To find the constant of variation, we compare the coefficients of x in both sides of the equation.

In this case, the coefficient of x is -10.
Therefore, the constant of variation is -10.
In conclusion, y varies directly with x, and the constant of variation is -10.

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A system of inequalities can have zero solutions, one solution, or infinitely many solutions, depending on the specific conditions and constraints of the inequalities involved.

A system of inequalities can have different numbers of solutions depending on the specific equations involved. Here are the possibilities:

1. No Solution: It's possible for a system of inequalities to have no solution, meaning there is no set of values that satisfies all the inequalities simultaneously. This happens when the inequalities are contradictory or when their solution sets don't overlap.

2. One Solution: In some cases, a system of inequalities can have a unique solution, where there is only one set of values that satisfies all the inequalities. This happens when the solution set for each inequality overlaps with the others in a specific way.

3. Infinite Solutions: Another possibility is that a system of inequalities can have infinitely many solutions. This occurs when the solution sets for the inequalities overlap completely or when the inequalities are equivalent.

Remember, the number of solutions can vary depending on the specific system of inequalities, so it's important to analyze each case individually.

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The sum of the arithmetic sequence is -468 (option D).

To find the sum of an arithmetic sequence, we can use the formula:

Sum = (n/2) * (first term + last term)

In this case, the first term of the sequence is 4, and the common difference between consecutive terms is -3. We need to find the last term of the sequence.

To find the last term, we can use the formula for the nth term of an arithmetic sequence:

last term = first term + (n - 1) * common difference

In this case, the last term is -56. We can use this information to find the number of terms (n) in the sequence:

-56 = 4 + (n - 1) * (-3)

-56 = 4 - 3n + 3

-56 - 4 + 3 = -3n

-53 = -3n

n = -53 / -3 = 17.67

Since the number of terms should be a whole number, we round up to the nearest whole number and get n = 18.

Now, we can find the sum of the arithmetic sequence:

Sum = (18/2) * (4 + (-56))

Sum = 9 * (-52)

Sum = -468

Therefore, the sum of the arithmetic sequence is -468 (option D).

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To approximate the integral ∫₀¹₂ y cos(y) dy using the trapezoidal rule, the midpoint rule, and Simpson's rule with the specified value of n, you need to divide the interval [0, 12] into n subintervals of equal width.

The formulas for each method are as follows:

Trapezoidal Rule:
Approximation = h/2 * [f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(xₙ₋₁) + f(xₙ)]
where h = (b - a)/n, x₀ = a, xₙ = b, and f(xᵢ) represents the value of the function at the midpoint of each subinterval.

Midpoint Rule:
Approximation = h * [f(x₀ + h/2) + f(x₁ + h/2) + ... + f(xₙ₋₁ + h/2)]
where h = (b - a)/n and xᵢ represents the left endpoint of each subinterval.

Simpson's Rule:
Approximation = h/3 * [f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + 4f(xₙ₋₁) + f(xₙ)]
where h = (b - a)/n, x₀ = a, xₙ = b, and f(xᵢ) represents the value of the function at each endpoint and midpoint of each subinterval.

Remember to round your answers to six decimal places.

In conclusion, to approximate the integral 12 ₀ y cos(y) dy using the trapezoidal rule, the midpoint rule, and Simpson's rule, divide the interval [0, 12] into n subintervals of equal width and apply the respective formulas mentioned above.

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The perimeter of the octagon is 16 units and the area is approximately 15.31 square units.

To find the perimeter and area of the regular octagon circumscribed about the circle with center Q(3,-1) and point X(1,-3), we need to determine the side length of the octagon.

Using the distance formula, we can find the distance between Q and X:

d(QX) = [tex]sqrt((1-3)^2 + (-3-(-1))^2)[/tex]

= [tex]sqrt((-2)^2 + (-2)^2)[/tex]

= [tex]sqrt(4 + 4)[/tex]

= [tex]sqrt(8)[/tex]

= 2sqrt(2)

Since the octagon is regular, all sides are equal. Therefore, the side length of the octagon is equal to d(QX) divided by sqrt(2):

side length =[tex](2sqrt(2)) / sqrt(2)[/tex]

= 2

The perimeter of the octagon is given by multiplying the side length by the number of sides:

perimeter = 8 * 2

= 16

To find the area of the octagon, we can use the formula:

area = [tex](2 * side length^2) * (1 + sqrt(2))[/tex]

= [tex](2 * 2^2) * (1 + sqrt(2))[/tex]

= [tex]8 * (1 + sqrt(2))[/tex]

≈ 15.31 (rounded to the nearest tenth)

The perimeter of the octagon is 16 units and the area is approximately 15.31 square units.

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Answer:

55 hours

Step-by-step explanation:

We can write an equation:

800=12x+150

And we can solve for x this way:

800=12x+150

subtract 150 from both sides

650=12x

divide both sides by 12

54.1666...=x

So, she will need to work 55 hours to get a new phone.  Unless the job that she works at pays her for half hour shifts, she needs to work 55 hours so she can buy the new phone.  She will have a little extra money left over too.

To show that two sides of one triangle are proportional to two corresponding sides of another triangle, with the included corresponding angles being congruent, you can use the Side-Side-Side (SSS) similarity criterion.

The SSS similarity criterion states that if the corresponding sides of two triangles are proportional and their corresponding angles are congruent, then the triangles are similar.
To prove this, follow these steps:

1. Given two triangles, let's call them triangle ABC and triangle DEF.
2. Identify two corresponding sides in each triangle that you want to show are proportional. Let's say AB and DE.
3. Also, identify the corresponding included angles, which are the angles formed by the corresponding sides. Let's say angle BAC and angle EDF.
4. Using the given information, state that AB/DE = BC/EF.
5. Now, prove that angle BAC = angle EDF. You can do this by showing that the two angles have the same measure or that they are congruent.
6. Once you have established that AB/DE = BC/EF and angle BAC = angle EDF, you can conclude that triangle ABC is similar to triangle DEF using the SSS similarity criterion.
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To use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field F across the surface S, we need to follow these steps:

1. Find the curl of the field F:
  The curl of F is given by ∇ × F, where ∇ is the del operator. In this case, F = 2yi + (5-3x)j + (z^2-2)k.

  ∇ × F = (d/dx, d/dy, d/dz) × (2yi + (5-3x)j + (z^2-2)k)
         = (0, 0, -3)

2. Determine the surface S and its orientation:
  The surface S is not specified in the question. Please provide the details of the surface S.

3. Calculate the flux of the curl of F across the surface S:
  Once we have the surface S and its orientation, we can evaluate the surface integral of the curl of F across S. The surface integral is given by the formula:

  ∬(curl F) · dS

  where dS represents the differential area vector on the surface S.

  Without knowing the details of the surface S, we cannot proceed with the calculation.

In conclusion, to calculate the flux of the curl of the field F across the surface S in the direction away from the origin, we need the specifics of the surface S. Please provide the necessary information so that we can proceed with the calculation.

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The set of values that represents a function is: (6, -5) (7, -3) (8, -1) (9, 1).

A set of values is considered a function if each input (x-value) is associated with only one output (y-value). Let's examine the given sets of values:

1. (2, -2) (5, 9) (5, -7) (1, 4)

  In this set, the x-value 5 is associated with two different y-values (-7 and 9). Therefore, this set of values is not a function.

2. (6, -5) (7, -3) (8, -1) (9, 1)

  Each x-value in this set is associated with a unique y-value. There are no repeated x-values, so this set of values is a function.

3. (3, 4) (4, -3) (7, 4) (3, 8)

  The x-value 3 is associated with two different y-values (4 and 8). Therefore, this set of values is not a function.

4. (9, 5) (10, 5) (9, -5) (10, -5)

  Each x-value in this set is associated with a unique y-value. There are no repeated x-values, so this set of values is a function.

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A parallelogram is a quadrilateral with two pairs of parallel sides. Here are the key properties of the sides, angles, and diagonals of a parallelogram:

1. Sides: The opposite sides of a parallelogram are congruent, which means they have the same length. This is due to the parallel nature of the sides.
2. Angles: The opposite angles of a parallelogram are also congruent. Additionally, the consecutive angles (adjacent angles that share a side) are supplementary, meaning they add up to 180 degrees.
3. Diagonals: The diagonals of a parallelogram bisect each other, meaning they divide each other into two equal parts. This property holds true for both the longer and shorter diagonals.
In summary, a parallelogram has congruent opposite sides and angles. The consecutive angles are supplementary, and the diagonals bisect each other. These properties are essential for understanding the fundamental characteristics of parallelograms.

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Comparing the total areas, we find that the second neighbor's party has more pizza in terms of area, with a total of 324π square inches compared to the first neighbor's party, which has a total of 320π square inches.

To determine which party has more pizza in terms of area, we need to calculate the total area of pizzas ordered by each neighbor.

First, let's calculate the area of a large pizza with a diameter of 16 inches. The formula for the area of a circle is A = πr^2, where A is the area and r is the radius. The radius of a 16-inch diameter pizza is half of the diameter, which is 8 inches.

So, the area of each large pizza is A = π(8 inches) ^2 = 64π square inches.

The first neighbor ordered 5 large pizzas, so the total area of pizzas for their party is 5 * 64π = 320π square inches.

Next, let's calculate the area of a small pizza with a diameter of 12 inches. Using the same formula, the radius of a 12-inch diameter pizza is 6 inches.

Thus, the area of each small pizza is A = π(6 inches)^2 = 36π square inches.

The second neighbor ordered 9 small pizzas, so the total area of pizzas for their party is 9 * 36π = 324π square inches.

Comparing the total areas, we find that the second neighbor's party has more pizza in terms of area, with a total of 324π square inches compared to the first neighbor's party, which has a total of 320π square inches.

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The probability that the mean annual snowfall during 25 randomly picked years will exceed a certain value, we need to calculate the z-score and look it up in the z-table to find the corresponding probability.

To find the probability that the mean annual snowfall during 25 randomly picked years will exceed a certain value, we need to use the properties of the normal distribution. Given that the amount of snowfall is normally distributed with a mean and a standard deviation, we can use the Central Limit Theorem.

The Central Limit Theorem states that if we have a sufficiently large sample size (in this case, 25 years), the distribution of the sample means will be approximately normal regardless of the shape of the population distribution.

To find the probability, we need to convert the mean annual snowfall into a standard score (also known as a z-score) using the formula:

z = (X - μ) / (σ / √(n)), where X is the value we want to find the probability for, μ is the mean, σ is the standard deviation, and n is the sample size.

Once we have the z-score, we can look it up in the z-table to find the corresponding probability. The probability represents the area under the normal distribution curve to the right of the z-score.

In conclusion, to find the probability that the mean annual snowfall during 25 randomly picked years will exceed a certain value, we need to calculate the z-score and look it up in the z-table to find the corresponding probability.

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In this problem, we are given the speed of two cyclists. Let's assume the speed of the slower cyclist to be x and the faster cyclist to be y. The two cyclists are moving towards each other, so the distance between them reduces with time. At the beginning, the distance between them is 105 miles, and at the end, it reduces to zero. Thus, we can say that the sum of the distances traveled by both cyclists is equal to the distance between them at the beginning.

This can be written as an equation: x t + y t = 105, where t is the time taken to meet each other. Since we have two unknowns x and y and only one equation, we cannot solve for both. However, we know that one cyclist is faster than the other, so y > x. We can use this fact to solve the problem.

We can isolate t by rewriting the above equation: x t + y t = 105, which gives us t = 105/(x + y). As the two cyclists meet each other in t hours, we can say that the slower cyclist covers a distance of xt, and the faster cyclist covers a distance of yt in this time. We know that the distance each cyclist covers is equal to their speed multiplied by the time. Thus, we can write: xt = 105/(x + y) and yt = 105/(x + y).

We can substitute these values of xt and yt in the equation x t + y t = 105, which gives us y x = 105. We can substitute x = y - r to get (y - r) y = 105. Simplifying this quadratic equation, we get y² - ry = 105. Solving this equation, we get y = 15 (since y > x, we take the positive root). We can find r by substituting y = 15 and x = y - r in the equation x t + y t = 105, which gives us r = 3.

Therefore, the speed of the slower cyclist is 12 mph, and the speed of the faster cyclist is 15 mph.

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The volume of air added to the balloon is approximately 386/3 cubic units.

To find the volume of air added to the balloon, we can use the formula for the volume of a sphere: V = (4/3)πr³.

First, we need to find the radius of the balloon before and after inflation. The formula for the circumference of a sphere is C = 2πr.

Given that the initial circumference is about 14 inches, we can solve for the initial radius:
14 = 2πr
r ≈ 14/(2π) ≈ 7/(π)

Similarly, for the final circumference of about 18 inches:
18 = 2πr
r ≈ 18/(2π) ≈ 9/(π)

Now that we have the initial and final radii, we can calculate the initial and final volumes:
Initial volume = (4/3)π(7/(π))³ = (4/3)π(343/(π³)) ≈ 343/3 cubic units
Final volume = (4/3)π(9/(π))³ = (4/3)π(729/(π³)) ≈ 729/3 cubic units

To find the volume of air added, we subtract the initial volume from the final volume:
Volume of air added = Final volume - Initial volume = (729/3) - (343/3) = 386/3 cubic units.

So, approximately 386/3 cubic units of air was added to the balloon.
The volume of air added to the balloon is approximately 386/3 cubic units.

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The discriminant is equal to 0, the equation has only one real solution.

To evaluate the discriminant of the equation -4x² + 20x - 25 = 0, we can use the formula Δ = b² - 4ac, where a, b, and c are the coefficients of the quadratic equation in the form ax² + bx + c = 0.

For the given equation, a = -4, b = 20, and c = -25. Substituting these values into the discriminant formula, we get Δ = (20)² - 4(-4)(-25).

Simplifying further, Δ = 400 - 400 = 0.

Since the discriminant is equal to 0, the equation has only one real solution.

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The actual length of the movie Jimmy went to see on Tuesday was 144 minutes.

Let's solve the problem step by step:

Step 1: Calculate the additional length of the movie.

The movie was 20% longer than what Jimmy thought. To find the additional length, we need to calculate 20% of the movie's length that Jimmy initially thought.

Additional length = 20% of the length Jimmy initially thought

Step 2: Calculate the actual length of the movie.

To find the actual length of the movie, we add the additional length to the length Jimmy initially thought.

Actual length = Length Jimmy initially thought + Additional length

Now let's calculate the additional length and the actual length using the given information:

Length Jimmy initially thought = 120 minutes

Step 1: Additional length

Additional length = 20% of 120 minutes

= (20/100) * 120

= 24 minutes

Step 2: Actual length

Actual length = Length Jimmy initially thought + Additional length

= 120 minutes + 24 minutes

= 144 minutes

Therefore, the actual length of the movie Jimmy went to see on Tuesday was 144 minutes.

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