Session 3
(Calculator)
David just bought six more baseball cards. The new baseball cards represent 30% of
David's special edition baseball card collection.)
Number of Baseball Cards
6
++
0
+
25 30
+
50
+
75
?
+
100
What is the total number of cards in David's baseball card collection?
Enter your answer in the box.

a) The relation set D is {(0,4), (0,5), (0,7), (3,6), (3,9), (4,4), (4,8), (4,12), (5,5), (5,10), (5,15), (7,7), (7,14)}. The relation diagram with arrows can be drawn as follows:

0 → 4, 5, 7
3 → 6, 9
4 → 4, 8, 12
5 → 5, 10, 15
7 → 7, 14


b) The relation set D-1 is {(4,0), (5,0), (7,0), (6,3), (9,3), (4,4), (8,4), (12,4), (5,5), (10,5), (15,5), (7,7), (14,7)}. The relation diagram with arrows can be drawn as follows:

4 → 0, 4, 8, 12
5 → 0, 5, 10, 15
7 → 0, 7, 14
6 → 3
9 → 3
8 → 4
12 → 4
10 → 5
15 → 5
14 → 7

a) The relation set D consists of pairs (x, y) such that x ∈ A and y ∈ B, and x divides y. D = {(0, 4), (0, 5), (0, 6), (0, 7), (0, 8), (0, 9), (0, 10), (0, 11), (3, 6), (3, 9), (4, 4), (4, 8), (5, 5), (5, 10), (7, 7)}. In the relation diagram, draw arrows from elements of A to elements of B according to these pairs.

b) The inverse relation set D⁻¹ consists of pairs (y, x) such that x ∈ A and y ∈ B, and x divides y. D⁻¹ = {(4, 0), (5, 0), (6, 0), (7, 0), (8, 0), (9, 0), (10, 0), (11, 0), (6, 3), (9, 3), (4, 4), (8, 4), (5, 5), (10, 5), (7, 7)}. In the relation diagram, draw arrows from elements of B to elements of A according to these pairs.

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Therefore, the margin of error for a 95% confidence interval for the population mean of AAA batteries (with a population standard deviation of 0.5 hours and a population mean of 9 hours) based on a sample size of 25 batteries is 0.196 hours.

What is margin error?

Margin of error is a statistical term that refers to the amount of error that is allowed in a survey or poll's results. It is a measure of the precision of the results and indicates the range within which the true value is likely to fall.

In statistical terms, the margin of error is calculated as a percentage of the total number of respondents in a survey, and it takes into account factors such as the size of the sample, the level of confidence desired, and the variability in the responses.

For example, if a poll has a margin of error of +/- 3%, it means that the actual value of the result could be up to 3% higher or 3% lower than the reported value in the survey.

The formula for the margin of error for a 95% confidence interval is:

Margin of error = z * (σ / [tex]\sqrt(n)[/tex])

Where:

z = the z-score for the desired confidence level (95% confidence level corresponds to z = 1.96)

σ = the population standard deviation

n = the sample size

Substituting the given values into the formula:

Margin of error = 1.96 * (0.5 /[tex]\sqrt(25)[/tex])

Margin of error = 1.96 * (0.5 / 5)

Margin of error = 1.96 * 0.1

Margin of error = 0.196

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The correct answer is (c) 0. The mean of the t-distribution is always 0.

The t-distribution is a probability distribution that is used to test hypotheses about the population mean when the sample size is small and the population standard deviation is unknown. The shape of the t-distribution depends on the degrees of freedom (df), which is equal to the sample size minus one.

Although the t-distribution changes shape as the degrees of freedom change, the center of the distribution is always at zero. Therefore, the mean of the t-distribution is always zero, regardless of the degrees of freedom or any other factors.

Therefore, the correct answer is (c) 0.

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The single transformation which this student performed include the following: a reflection over the y-axis.

In Geometry, a reflection over or across the y-axis or line x = 0 is represented and modeled by this transformation rule (x, y) → (-x, y).

By applying a reflection over the y-axis to the coordinate of the given triangle PQR, we have the following coordinates:

(x, y)                             →                 (-x, y).

Coordinate P = (-1, 1)   →  Coordinate J' = (-(-1), 1) = (1, 1).

Coordinate Q = (-4, 1)   →  Coordinate K' = (-(-4), 1) = (4, 1).

Coordinate R = (-4, 4)   →  Coordinate L' = (-(-4), 4) = (4, 4).

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Answer:

B. 17.1°

Step-by-step explanation:

Given that triangle ABC has a right angle at C, BC = 4 units and AC = 13 units.

We can use the Pythagorean theorem to find the length of AB, which is the hypotenuse of the right triangle:

AB² = AC² + BC²

AB² = 13² + 4²

AB² = 169 + 16

AB² = 185

AB = sqrt(185)

Now, to find angle A, we can use the sine function:

sin(A) = opposite/hypotenuse

sin(A) = BC/AB

sin(A) = 4/sqrt(185)

A = sin⁻¹(4/sqrt(185))

Using a calculator, we can find that:

A ≈ 17.10 degrees

Answer:

B. 17.1°

Step-by-step explanation:

Given that triangle ABC has a right angle at C, BC = 4 units and AC = 13 units.

We can use the Pythagorean theorem to find the length of AB, which is the hypotenuse of the right triangle:

AB² = AC² + BC²

AB² = 13² + 4²

AB² = 169 + 16

AB² = 185

AB = sqrt(185)

Now, to find angle A, we can use the sine function:

sin(A) = opposite/hypotenuse

sin(A) = BC/AB

sin(A) = 4/sqrt(185)

A = sin⁻¹(4/sqrt(185))

Using a calculator, we can find that:

A ≈ 17.10 degrees

The area of the triangle is 5 square units. None of the given options match our answer, so there may be an error in the question or the answer choices.

To determine the area of a triangle with vertices defined by the given points A(2,1), B(3,6), and C(6,2), we can use the formula for the area of a triangle:

Area = 1/2 * base * height

where the base is the distance between two vertices and the height is the perpendicular distance from the third vertex to the base. We can choose any two vertices as the base, so let's choose AB as the base.

The distance between A and B is:

√((3-2)^2 + (6-1)^2) = √(26)

To find the height, we need to find the equation of the line passing through C and perpendicular to AB. The slope of AB is (6-1)/(3-2) = 5, so the slope of the perpendicular line is -1/5. We can use the point-slope form to find the equation of the line:

y - 2 = (-1/5)(x - 6)
y = (-1/5)x + (32/5)

To find the height, we need to find the distance from point A to this line. We can use the formula for the distance from a point to a line:

distance = |Ax + By + C| / √(A² + B²)

where A, B, and C are the coefficients of the line in the standard form Ax + By + C = 0. Plugging in the values, we get:

distance = |2*(-1/5) + 1*1 + (32/5)| / √((-1/5)² + 1²)
distance = 10/√(26)

Now we can plug in the values into the formula for the area:

Area = 1/2 * √(26) * (10/√(26))
Area = 5

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Answer:

2 1/3

Step-by-step explanation:

V = LWH

V = 1 3/4 ft × 2/3 ft × 2 ft

V = 7/4 × 2/3 × 2/1 ft³

V = 28/12 ft³

V = 7/3 ft³

V = 2 1/3 ft³

The solution of inequality is -9 < x < 12.

The correct option is B.

We have,

-2 < x/3 + 1 < 5

Now, solving the inequation in parts as

-2 < x/3 + 1

-2 - 1 < x/3

-3 < x/3

-9 < x

and, x/3 + 1 < 5

x/3 < 4

x <12

Thus, the solution of inequality is -9 < x < 12.

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The slope of the parametric curve x=-4t^2-4, y=6t^3 at the point corresponding to t is -9t/4.

To find the slope of the parametric curve x=-4t^2-4, y=6t^3 at the point corresponding to t, follow these steps:
1.  Find the derivatives of both x and y with respect to t:
   dx/dt = -8t
   dy/dt = 18t^2
2. The slope of the parametric curve is the ratio of the derivatives, dy/dx.

    To find this, divide dy/dt by dx/dt:

    dy/dx = (dy/dt) / (dx/dt)

              = (18t^2) / (-8t)
3. Simplify the expression:
   dy/dx = -9t / 4
So, the slope of the parametric curve x=-4t^2-4, y=6t^3 at the point corresponding to t is -9t/4.

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(a) P{L1 = 0}: Since L1 is the last Poisson event time before m, and m is uniformly distributed on [0, 1], we have:

P{L1 = 0} = P{no Poisson events in [0, m]} = e^(-λm)

(b) P{L1 < x}, 0 < x < m: We have:

P{L1 < x} = P{there is at least one Poisson event in [0, x]} = 1 - P{no Poisson events in [0, x]} = 1 - e^(-λx)

(c) P{R1 = 1}: Since R1 is the first Poisson event time after m, we have:

P{R1 = 1} = P{no Poisson events in [m, 1]} = e^(-λ(1-m))

(d) P{R1 = x}, m < x < 1: We have:

P{R1 = x} = P{there is no Poisson event in [m, x]} * P{there is at least one Poisson event in [x, 1]}

= e^(-λ(x-m)) * (1 - e^(-λ(1-x)))

(e) E[R-L]: We have:

E[R-L] = E[(R1-L1) + (R2-L2) + ... + (Rn-Ln)]

= E[R1-L1] + E[R2-L2] + ... + E[Rn-Ln] (by linearity of expectation)

= nE[R1-L1] (since the n copies are identical)

To find E[R1-L1], note that R1-L1 represents the length of the fragment that contains the marked position. This length is distributed as an exponential random variable with parameter λ, since it is the time until the next Poisson event in a Poisson process with rate λ. Therefore:

E[R1-L1] = 1/λ

Thus, we have:

E[R-L] = n/λ

This means that as n (the number of copies) becomes large, the expected length of the interval containing the marked position becomes smaller and smaller, converging to 0 as n approaches infinity.

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The quadratic value equation with zeroes at -11 is f(x) = a(x + 11)²

Given data ,

A quadratic function that has -11 as its only zero can be written in the form:

f(x) = a(x - r)²

where "a" is a non-zero constant and "r" is the zero of the function, in this case, -11.

On simplifying the equation , we get

f(x) = a(x - (-11))²

f(x) = a(x + 11)²

Hence , any quadratic function of the form f(x) = a(x + 11)^2, where "a" is a non-zero constant, will have -11 as its only zero

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The statement is false because in exponential growth, the time required to increase by a factor of 10 actually increases as the quantity gets larger.

We have,

In exponential growth, the rate of increase becomes progressively faster as the quantity grows.

This means that the time it takes to increase by a factor of 10 will also increase. For example, if it takes 1 year to increase from 1 to 10, it may take 2 years to increase from 10 to 100, and 3 years to increase from 100 to 1000.

The time required to achieve a factor of 10 increase will depend on the specific growth rate and initial quantity.

Therefore,

The time required to increase by a factor of 10 does not remain constant for all time in exponential growth. It increases as the quantity grows larger.

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The equation 4 = a1 + b2 + c*3. Therefore, 4 cannot be written as a linear combination of {1, 2, 3}.

4 cannot be written as a linear combination of {1, 2, 3}. To show this, we can assume the opposite and try to find coefficients that satisfy the equation 4 = a1 + b2 + c*3, where a, b, and c are constants.

Subtracting 2 from both sides, we get:

2 = a*(-1) + b0 + c1

This is a system of two equations with three variables, which does not have a unique solution. We can solve for one of the variables in terms of the other two, for example:

a = 2 - c

b = any value

c = any value

This means that there are infinitely many solutions, and we cannot find a unique combination of a, b, and c that satisfies the equation 4 = a1 + b2 + c*3. Therefore, 4 cannot be written as a linear combination of {1, 2, 3}.

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In conclusion, each yi (i = 1, ..., n) has a Poisson distribution with mean μ, and their PMFs follow the below expression.

Hi! The given terms are y1, ..., yn, which are independent Poisson random variables each with mean μ. To determine the distribution for y1, ..., yn, we consider their properties.

Since y1, ..., yn are independent Poisson random variables, each of them follows a Poisson distribution with the same mean μ. The probability mass function (PMF) for each yi (where i = 1, ..., n) can be expressed as:

[tex]P(y_i = k) = (e^{-u} * \frac{(u^k))} { k!} , for k = 0, 1, 2, ...[/tex]

Here, e is the base of the natural logarithm, and k! denotes the factorial of k.

In conclusion, each yi (i = 1, ..., n) has a Poisson distribution with mean μ, and their PMFs follow the above expression.

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The option to the above question is a circle graph titled New York City visitor's transportation, with five sections labelled walk 40 per cent, bicycle 8 per cent, car service 15 per cent, bus 10 per cent, and subway 27 per cent.

A circle is a geometric shape that consists of a set of points in a plane that are equidistant from a fixed point called the centre. The distance from the centre to any point on the circle is called the radius, and the distance across the circle passing through the centre is called the diameter.

According to the given information:

The correct circle graph that represents the data in the table would be:

circle graph titled New York City visitor's transportation, with five sections labelled walk 40 per cent, bicycle 8 per cent, car service 15 per cent, bus 10 per cent, and subway 27 per cent.

This is because the percentages shown in this circle graph match the data given in the table for each type of transportation. Specifically, the circle graph shows that 40% of visitors walked, 8% used bicycles, 15% used car service, 10% used the bus, and 27% used the subway, which aligns with the numbers provided in the table.

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Answer:It’s B i got it correct on the quiz

Step-by-step explanation:

The probability of getting at least one three of a kind in 20 hands of poker is approximately 49%.

The Poisson approximation is a method of estimating the probability of a rare event. The formula used is P(x) = (e^lambda * lambdaˣ) / x! where lambda is the average number of occurrences of the event.

In this case, we are looking for the probability of getting at least one three of a kind in 20 hands of poker.

The probability of getting a three of a kind in one hand is 1/50.

Therefore, the average number of occurrences of a three of a kind in 20 hands is (20 x 1/50) = 0.4.

Using the Poisson approximation, we get P(x) = (e⁰.⁴ x (0.4)ˣ) / x!

In this case, x = 1, so

P(x) = (e⁰.⁴ x (0.4)¹) / 1

= 0.49

= 49%.

Therefore, the probability of getting at least one three of a kind in 20 hands of poker is approximately 49%.

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Answer:

Step-by-step explanation:

1=a

Answer:

Step-by-step explanation:

Identity means to get itself

For multiplication,

if you  multiplied a number by 0, you would not get that number again, you would get 0.  so multiplying by 0 is not and identity

ex.  (a)(0)=0   does not = a so this is not an identity

if you multiplied a number by 1, yes that would be an identity because any number times 1 is itself

ex. a(1)=a    multiplied by 1 the number is itself, so yes this is an identity.

The circle that contains the point (-2, 8) and has a center at (4, 0) is given by the equation (x - 4)² + y² = 100

The standard form equation of a circle with center (h, k) and radius r is:

(x - h)² + (y - k)² = r²

Given that: the circle that contains the point (-2, 8) and has a center at (4, 0).

The distance between the center of a circle and any point on the circle is constant and is called the radius of the circle.

TherHence, to find the circle that contains the point (-2, 8) and has a center at (4, 0), we need to find the distance between these two points and use that as the radius of the circle.

The distance between two points (x1, y1) and (x2, y2) is given by the distance formula:

d = √((x2 - x1)² + (y2 - y1)²)

d = √((4 - (-2))² + (0 - 8)²)

= √(6² + (-8)²)

= √(100)

= 10

Hence, the distance between the center of the circle at (4, 0) and the point (-2, 8) is 10 units.

Therefore, the radius of the circle is 10 units.

The equation of a circle with center (h,k) and radius r is given by:

(x - h)² + (y - k)² = r²

In this case, the center is (4, 0) and the radius is 10, so the equation of the circle is:

(x - 4)² + y² = 10²

(x - 4)² + y² = 100

Therefore, the equation of the circle is (x - 4)² + y² = 100.

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ANSWER

480 cm^2

Step-by-step explanation:

divide it into 3 blocks

Answer:

a) Yes No Total

Students 30 15 45

Teachers 20 10 30

Total 50 25 75

b) 30/75 = 2/5 = 40% of the people surveyed were teachers.

c) 20/75 = 4/15 = 26.7% of the people surveyed were teachers who wanted a later start time.

Each group of four binary digits can be represented by a single hexadecimal digit. The hexadecimal notation of (0111 0111 0111 0111)₂ is (7777)₁₆.

To convert the binary expansion of (0111 0111 0111 0111)2 to hexadecimal, we first group the digits into groups of four starting from the right:
(0111 0111 0111 0111)2 = (7 7 7 7)16
Each group of four binary digits can be represented by a single hexadecimal digit. In this case, each group of four binary digits represents the hexadecimal digit 7. Therefore, the hexadecimal notation of (0111 0111 0111 0111)2 is (7777)16.
To convert the binary expansion (0111 0111 0111 0111)₂ to a hexadecimal expansion, you can group the binary digits into sets of four starting from the right and then convert each group to its corresponding hexadecimal value. Here's the process:
1. Group the binary digits: (0111) (0111) (0111) (0111)
2. Convert each group to hexadecimal:
  - (0111)₂ = 7₁₆
  - (0111)₂ = 7₁₆
  - (0111)₂ = 7₁₆
  - (0111)₂ = 7₁₆
Your answer: The hexadecimal notation of (0111 0111 0111 0111)₂ is (7777)₁₆.

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Answer:

LO=6

Step-by-step explanation:

MO/KO=NO/LO

5/10=3/LO

5LO=30

LO=6cm

Triangle DEF is not congruent to triangle ABC

Yes, triangle GHI is congruent to triangle ABC the reason is SAS

The Angle-Side-Angle (ASA) theorem is a geometry mathematical principle that establishes the congruence of triangles.

More specifically, this theorem notes that if two angles and their included side on one triangle are equal in measure to the corresponding two angles and included side on another triangle, then both triangles are said to be congruent.

Since ASA relies heavily upon the matching of angle size and side length, it serves as an essential tool for geometric proofs and thorough analyses.

The corresponding angles are

52.4 and 45.5

The included side is 5cm

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For any integer n, [tex]n^2[/tex] is congruent to 0 or 1 mod 4. This statement is true.

Let's first consider the possible remainders of an integer when divided by 4. There are four possibilities: 0, 1, 2, or 3.

If we square any integer, we get an even number if the original integer is even (i.e., has remainder 0 or 2 when divided by 4), and we get an odd number if the original integer is odd (i.e., has remainder 1 or 3 when divided by 4).

Now, let's consider the possible remainders of [tex]n^2[/tex] when divided by 4:

If n has remainder 0 when divided by 4 (i.e., n is even), then [tex]n^2[/tex] has remainder 0 when divided by 4, since the square of any even number is divisible by 4. So,[tex]n^2[/tex] is congruent to 0 mod 4.

If n has remainder 1 when divided by 4 (i.e., n is odd), then [tex]n^2[/tex] has remainder 1 when divided by 4, since the square of any odd number leaves a remainder of 1 when divided by 4. So, [tex]n^2[/tex] is congruent to 1 mod 4.

If n has remainder 2 when divided by 4 (i.e., n is even), then [tex]n^2[/tex] has remainder 0 when divided by 4, since the square of any even number is divisible by 4. So, [tex]n^2[/tex] is congruent to 0 mod 4.

If n has remainder 3 when divided by 4 (i.e., n is odd), then [tex]n^2[/tex] has remainder 1 when divided by 4, since the square of any odd number leaves a remainder of 1 when divided by 4. So, [tex]n^2[/tex] is congruent to 1 mod 4.

Therefore, we have shown that for any integer n, [tex]n^2[/tex] is congruent to 0 or 1 mod 4.

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Answer:

The distance the players must run is [tex]150 m[/tex]

Step-by-step explanation:

The distance that the players must run diagonally from one corner of the soccer field to the inverse corner can be found by using the Pythagorean hypothesis,

The Pythagorean hypothesis may be a scientific guideline that relates to the sides of a right triangle. It states that the square of the length of the hypotenuse (the longest side of the triangle) is break even with to the whole of the squares of the lengths of the other two sides.  

The length of the soccer field is 120 m long.(given)

The width of the soccer field is 90 m (given)

and width of the soccer field shape the two legs of the right triangle, and the corner-to-corner distance is the hypotenuse. Hence, we can utilize the Pythagorean theorem as takes after:

Distance = [tex]\sqrt{(length^{2} } + width^{2}[/tex]

                [tex]= \sqrt{120^{2} + 90^{2} }[/tex]

                = ([tex]\sqrt{(14400 + 8100)}[/tex]

                = [tex]\sqrt{22500}[/tex]

                 [tex]= 150.00 meters[/tex]

Therefore, the distance the players must run is 150. m

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Answer:

Step-by-step explanation:

It is a right triangle.

Pythagorean Theorem can be used to find the sides.

IF the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.  The right angle is opposite the longest side.

72² + 8.5² = 72.5²

5184 + 72.25 = 5256.25

5256.25 = 5256.25

(a) f = 5xyz + 5x^2y/2 + C is a potential function for F.

(b) f = ye^(xz) + 9x^2y/2e^(xz) + C is a potential function for F.

To find a potential function f for a conservative vector field F, we need to find a scalar function f(x, y, z) such that the gradient of f is equal to F, i.e., ∇f = F.

(1) For F = 5yzi + 5xzj + 5xyk, we need to find f such that ∂f/∂x = 5yz, ∂f/∂y = 5xz, and ∂f/∂z = 5xy.

Integrating the first equation with respect to x gives f = 5xyz + g(y, z), where;

Differentiating this expression with respect to y and z and comparing with the other two equations, we find that;

g(y, z) = C + 5x^2y/2 and

f = 5xyz + 5x^2y/2 + C,

where;

Therefore, f = 5xyz + 5x^2y/2 + C is a potential function for F.

(2) For F = 9yze^(xz)i + 9exzj + 9xye^(xz)k, we need to find f such that;

Integrating the first equation with respect to x gives f = ye^(xz) + g(y, z),.

where;

Differentiating this expression with respect to y and z and comparing with the other two equations, we find that;

g(y, z) = C + 9x^2ye^(xz)/2 and

f = ye^(xz) + 9x^2y/2e^(xz) + C,

where;

Therefore, f = ye^(xz) + 9x^2y/2e^(xz) + C is a potential function for F.

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Answer:

40%

Step-by-step explanation:

boys can cycle = 49- 22 = 27

total boys = 27+ 41  = 68

% can cycle = 27/ 68 = 40%

Answer:

Step-by-step explanation: