Evaluate each determinant

As a result, there are 25,200 potential 2-person combinations that can be made up of a junior and a senior.

In mathematics, n is used to represent the factorial of a non-negative number, n! n less than or equal to and are said to be the product of all positive integers. For instance:

5! = 5 x 4 x 3 x 2 x 1 = 120

Several branches of mathematics, including algebra, mathematical analysis, and combinatorics, use the factorial function.

It is used to determine how many different ways there are to arrange given objects.

Using the formula for combinations, we can determine how many different 2-person combinations with a junior and a senior are conceivable.

nCr = n / r!(n-r)!

where r is the number of persons we wish to choose from (2), n is the total number of people (125 juniors and 100 seniors), and! stands for the factorial function.

Adding these values to the formula yields the following results:

(125 plus 100)C2

= (125 plus 100)! / 2!(125 plus 100-2)!

= (225)! / 2!(223)!

= (225 x 224 x 223!) / (2 x 1 x 223!)

= (225 x 224)! / 2

= 25,200

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Answer:

4x^2 + x + 3 + 4/x-1

Step-by-step explanation:

Answer:

The length of diagonal EG and FH in rectangle EFGH can be expressed as 8x - 1 and 3x + 9, respectively.

Step-by-step explanation:

In a rectangle, the diagonals are equal in length. Therefore, we can set 8x - 1 equal to 3x + 9 and solve for x:

8x - 1 = 3x + 9

5x = 10

x = 2

Now that we know x = 2, we can find the length of both diagonals:

EG = 8x - 1 = 8(2) - 1 = 15

FH = 3x + 9 = 3(2) + 9 = 15

Therefore, the length of both diagonals in rectangle EFGH is 15.

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The equation of the least-squares regression line based on these data is, y= 106.92 +2.044x where y =predicted height of child and x = child’s foot length.

To determine the equation of the least-squares regression line, we need to fit a linear model to the data using the method of least squares. The equation of the line can be written as:

height = intercept + slope * foot length

where the intercept and slope are the parameters we need to estimate from the data. The intercept represents the predicted height when foot length is 0, and the slope represents the change in height for each unit increase in foot length.

Substituting the values form the graph,

y= 106.92 +2.044x where y =predicted height of child and x = child’s foot length.

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--The complete question is, Can we predict the heights of school-aged children from foot length? below is computer output from a regression analysis of this relationship for 15 randomly-selected canadian children from 8 to 15 years old, along with a residual plot. the explanatory variable is each child's foot length (in centimeters), and the response variable is the child's height (in centimeters).

What is the equation of the least-squares regression line based on these data?  Define any parameters used.--

Answer:

x = 10

Step-by-step explanation:

using the cosine ratio in the right triangle and the exact value

cos60° = [tex]\frac{1}{2}[/tex] , then

cos60° = [tex]\frac{adjacent}{hypotenuse}[/tex] = [tex]\frac{5}{x}[/tex] = [tex]\frac{1}{2}[/tex] ( cross- multiply )

x = 5 × 2 = 10

Answer: The length, breadth, and height of the given object are 3x+y cm, 2x+2y cm, and 3y cm, respectively. To find the volume of this object, we need to multiply these three dimensions:

Volume = Length x Breadth x Height

= (3x+y) x (2x+2y) x (3y)

= 18x^2y + 27xy^2 + 6xy^2 + 6y^3

= 18x^2y + 33xy^2 + 6y^3

Therefore, the volume of the object is 18x^2y + 33xy^2 + 6y^3 cubic cm.

Step-by-step explanation:

You can follow the order of operations with all of the other operations in the equation and treat the operations in the expression separately. Option 2.

The order of operations is a set of rules that dictate the sequence in which mathematical operations should be performed in an expression. These rules are designed to ensure that the expression is evaluated correctly and consistently.

When interpreting an expression as a given quantity, you are essentially substituting a value for a variable in the expression. This does not contradict the order of operations because the rules still apply, but the variables in the expression are treated as known values rather than unknown variables.

For example, if you have the expression 2 + 3 × 4, following the order of operations, you would first perform the multiplication (3 × 4 = 12) and then add the result to 2 (2 + 12 = 14).

If you are asked to interpret this expression as a given quantity when x = 3, you would substitute 3 for x and get 2 + 3 × 4 = 14, which is the same result obtained by following the order of operations.

In short, interpreting an expression as a given quantity does not contradict the order of operations because the rules still apply, but the variables in the expression are treated as known values.

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The value of angle x in the intersecting chords is determined as  146⁰.

The value of angle x is determined by applying intercepting chord theorem for tangent angle at circumference of a circle.

The intercepting chord theorem, also known as the tangent chord theorem, states that when a tangent line intersects a chord of a circle at a point on the chord, then the measure of the angle formed by the tangent line and the chord is equal to half the measure of the intercepted arc (the arc that lies between the endpoints of the chord).

So if the tangent angle = 73⁰, the arc angle X = 2(73) = 146⁰

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Answer:

Step-by-step explanation:

So, first you add the four exam percentage which will be 314 then you multiply 80 and 5. We use 5 because there are total 5 exams which will give the mean of 80%. After you get the answer for 80x5 which will be 400. Then you do, 400 - 314 which will be 86. So, the answer is 86%. To figure out if that answer is right, You add all the scores percentage and divide that by 5 and you should get 80%.

To have an overall mean of 80% after five exams, the student needs to score 88% on the next exam.

To find the student's overall mean, we need to add up all the scores and divide by the number of exams.

We have 4 exams scores:84%, 71%, 86%, and 73%.

Therefore, the total of the four exam scores is:

84% + 71% + 86% + 73% = 314%

To find the overall mean after five exams, the total score after five exams should be 5 × 80% = 400%.

Therefore, to have an overall mean of 80%, the student must score (400 - 314)% on the fifth exam, which is equivalent to: 86%.

Hence, the student needs to score 86% on the next exam to have an overall mean of 80%.

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This result doesn't make sense in the context of the problem, so it's likely that there was a mistake in the problem setup or in the information given.

What is Algebraic expression ?

An algebraic expression is a mathematical phrase that can include numbers, variables, and operators (such as addition, subtraction, multiplication, and division), as well as grouping symbols like parentheses.

Let's assume that they both sold a total of $x.

Since Rita sold each box of cookies for $10, she must have sold x/10 boxes.

Likewise, since John sold each box for $8, he must have sold x/8 boxes.

We know that they both sold the same dollar amount, so we can set their two sales expressions equal to each other and solve for x:

x/10 = x/8

Multiplying both sides by the least common multiple of 10 and 8, which is 40:

4x = 5x

Subtracting 4x from both sides:

x = 0

Therefore, This result doesn't make sense in the context of the problem, so it's likely that there was a mistake in the problem setup or in the information given.

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The given triangles are congruent by AAS congruence. The solution has been obtained by using the concept of congruent triangles.

What are congruent triangles?

Triangles which have the same size and the same shape are said to be congruent. This implies that the respective sides and angles of the triangles are both equal.

We are given two triangles as ABC and CDE in which C is the midpoint of BD.

So, from this, we get

⇒ AC = EC                                 (Given)

⇒ ∠ABC = ∠CDE                       (Right Angle)

⇒ ∠ACB = ∠ECD                       (Common Angle)

Hence, the triangles are congruent by the AAS congruence.

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Answer:

3 more kids

Step-by-step explanation:

Answer:

3 kids can join in on the fun.

Step-by-step explanation:

Total number of kids who can jump on the trampoline = 6

Kids jumping currently = 3

Therefore, The number of kids who can join the fun will be the difference of the total number of kids who can jump on the trampoline and kids jumping currently.

= 6 - 3

= 3

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Answer: 75

Step-by-step explanation:

[tex]x[/tex]% x 3000000 = 2250000

[tex]x[/tex]/100 x 3000000 = 2250000

[tex]x[/tex] x 30000 = 2250000

[tex]x[/tex] = 2250000/30000

[tex]x[/tex] = 75

The area between the two curves will be around 17/6 square units.

Let's start by setting the two equations equal to each other:

|+3|−2 = 1−(1/3)|−2|

Case 1: +3≥0

In this case, the equation simplifies to:

+3−2 = 1−(1/3)|−2|

Simplifying further, we get:

= 1

Case 2: +3<0

In this case, the equation simplifies to:

−(+3)−2 = 1−(1/3)|−2|

Simplifying further, we get:

= −2

two curves intersect at =−2 and =1.

we need to integrate the difference between the two curves with respect to , from =−2 to =1:

∫−2¹ [(1−(1/3)|−2|)−(|+3|−2)] d

Simplifying the absolute values,

∫−2¹ [(1−(1/3)(−2))−(|+3|−2)] d

Next, we can break up the integral into two parts:

∫−2¹ [1−(1/3)(−2)] d − ∫−2¹ (|+3|−2) d

Simplifying, (7/3) + (1/2) = 17/6

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The formula for the surface area of a cylinder is:

S = 2πrh + 2πr²

To factor the right side of the formula, we can first take out the common factor of 2πr:

S = 2πr(h + r)

Therefore, the right side of the formula can be factored as 2πr(h + r).

Answer:

20736

Step-by-step explanation:

12*12*12*12=20736

Answer:

the mode is 2 In order 0,1,2,3,9,12,14

Step-by-step explanation:

The reason is the mode equals the middle and 2 is there twice and the 2 is in the middle

The required probability and average for the given arrival rate and service rate are,

Average time a car in the given system is 2.13 minutes.

Average number of cars in the given system is 0.852 cars.

Average number of cars waiting is 0.052 cars.

Average time a car spends waiting is 0.13 minutes.

Probability of no cars is 0.2.

Percentage of time the serving pharmacist is 80%.

Probability of exactly 2 cars is 0.64.

Arrival rate per minute λ = 4/10 = 0.4

And service rate per minute µ = 1/2

                                                  = 0.5 .

Average time a car is in the system is given by,

W = Wq + 1/µ

where Wq is the average time a car spends waiting in the queue.

1/µ is the average service time.

Using Little's Law, we have,

L = λW

where L is the average number of cars in the system.

Using the given values,

Wq

= (0.4^2)/(2×(1-0.4))

= 0.13 minutes

W

= Wq + 1/µ

= 0.13 + 2

= 2.13 minutes

Average time a car is in the system is 2.13 minutes.

Average number of cars in the system is,

L = λW

  = λWq + λ/µ

L = λWq + λ/µ

   = 0.4×0.13 + 0.4/0.5

   = 0.052 + 0.8

   = 0.852 cars

Average number of cars in the system is 0.852 cars.

Average number of cars waiting in the queue is,

Lq = λWq

    = 0.4×0.13

    = 0.052 cars

Average number of cars waiting in the queue is 0.052 cars.

Average time a car spends waiting in the queue is,

Wq = Lq/λ

     = 0.052/0.4

     = 0.13 minutes

Average time a car spends waiting in the queue is 0.13 minutes.

Probability that there are no cars at the window is,

P(0) = (1 - λ/µ)

      = (1 - 0.4/0.5)

      = 0.2

Probability that there are no cars at the window is 0.2.

Percentage of time the serving pharmacist is busy is ,

ρ = λ/µ

  = 0.4/0.5

  = 0.8

Percentage of time the serving pharmacist is busy is 80%.

Probability that there are exactly 2 cars in the system is,

P(2) = ((λ/µ)^2/(1-ρ)) × P(0)

where P(0) is the probability that there are no cars at the window.

P(2)

= ((0.4/0.5)^2/(1-0.8))×0.2

= 0.64

Probability that there are exactly 2 cars in the system is 0.64.

Therefore, the probability for the given situations are,

Average time is 2.13 minutes.

Average number of cars is 0.852 cars.

Average number of cars waiting in the queue is 0.052 cars.

Average time a car spends waiting in the queue is 0.13 minutes.

Probability of no cars at the window is 0.2.

Percentage of time the serving pharmacist is busy is 80%.

Probability that there are exactly 2 cars is 0.64.

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Answer:

-31/30

Step-by-step explanation:

-5/6 + 1/3 Times -3/5

-5/6 - 1/5

The average number of calls per day is approximately 5.298 calls.

To solve this problem, we'll first use the information given about the probability of receiving no calls (0.5%) and the Poisson distribution formula to find the average number of calls per day (λ).

Step 1: Convert the percentage of no calls into a decimal.
0.5% = 0.005

Step 2: Use the Poisson distribution formula for the probability of receiving no calls (k = 0).
P(X = 0) = (e^(-λ) * λ^0) / 0! = 0.005

Step 3: Simplify the equation.
(e^(-λ) * 1) / 1 = 0.005

Step 4: Solve for λ.
e^(-λ) = 0.005
-λ = ln(0.005)
λ = -ln(0.005)

Step 5: Calculate the average number of calls per day.
λ ≈ 5.298

So, the average number of calls per day is approximately 5.298 calls.

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About 0.331 grams of the substance is left after 180 days have passed.

The decay of iodine-131 can be modelled by the equation:

[tex]y(t) = y0 * e^(-kt)[/tex]

where y0 is the initial mass, t is time in days, and k is the decay constant.

We are given that y(7) = 6 grams, so we can plug in these values and solve for k:

[tex]6 = y0 * e^(-7k)\\y0 = 6 / e^(-7k)[/tex]

We are also given that 180 days have passed, so we can use the equation to find y(180):

[tex]y(180) = y0 * e^(-k*180)[/tex]

Substituting y0 from the previous equation:

[tex]y(180) = 6 / e^(-7k) * e^(-k*180)[/tex]

Simplifying:

[tex]y(180) = 6 * e^(-k*173)[/tex]

We need to find k in order to evaluate this expression. To do so, we can use the fact that the half-life of iodine-131 is about 8 days. This means that:

[tex]1/2 = e^{(-k*8)}[/tex]

Taking the natural logarithm of both sides:

[tex]ln(1/2) = -8k\\k = -ln(1/2) / 8\\k \approx 0.08664[/tex]

Substituting this value of k into the expression for y(180):

[tex]y(180) = 6 * e^(-0.08664*173)\\y(180) \approx 0.331 grams[/tex]

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THE GRAPH SHOWING (X=2,5) (Y=10,9) (Z=6,1) Thus, taking YZ as the base, the area of the triangle XYZ is 2√(98)square units.

The total area filled by a triangle's three sides in a two-dimensional plane is what is known as the triangle's area. The basic formula for calculating a triangle's area is A = 1/2× b× h, which equals half the product of the triangle's base and height.

We can use the following formula to determine a triangle's area given its coordinates:

Area is equal to half of |x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|.

where the triangle's vertices' coordinates are (x₁,y₁), (x₂,y₂), and (x₃,y₃).

As YZ serves as our base in this instance, we must first determine its length. When we apply the distance formula, we obtain:

√((1-9)2 + (6-1)2) Equals √YZ (98)

Finding the perpendicular distance from X to YZ will allow us to determine the height of the triangle now that we know YZ. This distance is equivalent to 2-6 = -4, which is the x-coordinate of X less the x-coordinate of Z.

As a result, the triangle's area is:

Area is equal to 1/2 × YZ × height

= 1/2×√(98)× |-4|

= 2 √ (98)

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Answer:

D

Step-by-step explanation:

Convert all of the fractions' denominators so that they are 8. This is because 4,2, and 8's least common multiple is 8.

Remember that when multiplying the denominator, we must also multiply the numerator. So:

[tex]\frac{1}{2} * \frac{4}{4} = \frac{4}{8} \\[/tex]

[tex]\frac{1}{4} *\frac{2}{2} =\frac{2}{8}[/tex]

Since we are asked to add all of them, we can do that easily since all of the fractions now share a common denominator.

[tex]\frac{2}{8} +\frac{4}{8} + \frac{1}{8} = \frac{7}{8}[/tex]

Therefore, the answer is D.

From the histogram, we can see that there are 4 outcomes in which three heads appear. Therefore, the probability that three heads will appear on any toss is 4/16 = 0.25.

Consider the toss of four coins. There are 24 possible outcomes of a single toss. Develop a histogram of the number of heads (one side of the coin) that can appear on any toss. Does it look like a normal distribution? Should this be expected? What is the probability that three heads will appear on any toss?Solution:When four coins are tossed, there are 16 possible outcomes.

Each outcome has an equal probability of 1/16 of occurring. A histogram of the number of heads that can appear on any toss is shown below:Based on the histogram, the probability of getting 0 heads or 4 heads is 0.375, while the probability of getting 1 head or 3 heads is 0.25, and the probability of getting 2 heads is 0.125.

The histogram does not look like a normal distribution. This is because a normal distribution has a bell-shaped curve, which is not the case with this histogram. This is because there are a finite number of possible outcomes, and the probabilities of these outcomes are not normally distributed.

Thus, the histogram is not expected to look like a normal distribution .The probability that three heads will appear on any toss is the sum of the probabilities of the outcomes in which three heads appear.

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Answer:

55/99

Step-by-step explanation:

To express 0.55 repeating as a fraction, we can use the following steps:

Let x = 0.55 repeating

Multiplying both sides by 100 to eliminate the repeating decimal gives:

100x = 55.55 repeating

Subtracting the left-hand side (100x) from the right-hand side (55.55 repeating) gives:

99x = 55

Dividing both sides by 99 yields:

x = 55/99

Therefore, 0.55 repeating can be expressed as the fraction 55/99.

The 95 % confidence interval based on the same data is 90 ± 30. Option b is right choice.

To calculate the 95% confidence interval based on the same data, we need to use the formula:

CI = [tex]\bar x[/tex] ± zα/2 x (σ / √n)

where:

[tex]\bar x[/tex] is the sample mean

zα/2 is the z-score corresponding to the desired confidence level (in this case, 95% confidence level, so zα/2 = 1.96)

σ is the population standard deviation (which is unknown, so we'll use the sample standard deviation instead)

n is the sample size

Given that the 90% confidence interval is 90 ± 30, we can conclude that:

[tex]\bar x[/tex] = 90

the margin of error = 30 / 1.645 (corresponding to the z-score for 90% confidence level) = 18.22

the sample standard deviation is unknown, so we cannot determine the exact value of σ.

Therefore, the 95% confidence interval can be calculated as follows:

CI = 90 ± 1.96 x (18.22 / √n)

To find the answer choice that could be the 95% confidence interval, we need to check which one contains the range of values that can be obtained using this formula for different sample sizes.

a. 90 ± 21: This range is too narrow to be a 95% confidence interval. It corresponds to a margin of error of

21 / 1.96 = 10.71

which is smaller than the margin of error for the 90% confidence interval (which was 30 / 1.645 = 18.22).

Therefore, this option is not correct.

b. 90 ± 30: This is the same range as the 90% confidence interval, which we already know is correct.

Therefore, this option is also correct.

c. 90 ± 39: This range is too wide to be a 95% confidence interval. It corresponds to a margin of error of

39 / 1.96 = 19.90

which is larger than the margin of error for the 90% confidence interval (which was 30 / 1.645 = 18.22).

Therefore, this option is not correct.

d. 90 ± 70: This range is also too wide to be a 95% confidence interval. It corresponds to a margin of error of 70 / 1.96 = 35.71, which is much larger than the margin of error for the 90% confidence interval (which was 30 / 1.645 = 18.22).

Therefore, this option is not correct.

Therefore, the correct answers are b. 90 ± 30.

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Question:

A 90% confidence interval for the mean μ of a population is computed from a random sample and is found to be 90 ± 30. Which of the following could be the 95 % confidence interval based on the same data?

a. 90 ± 21

b. 90 ± 30

c. 90 ± 39

d. 90 ± 70

Answer:

So Adam has eaten 1/6 of the pizza.  :)  ;)

Step-by-step explanation:

Assuming the pizza is cut into 8 equal pieces (which would each be 45 degrees of the total 360 degrees of the pizza), we can calculate how much of the pizza Adam has eaten with the given information.

First, we add up the angles of the two pieces Adam has eaten:

35 degrees + 25 degrees = 60 degrees

This means that Adam has eaten 60 degrees out of the total 360 degrees of the pizza. To convert this to a fraction, we divide 60 by 360:

60 / 360 = 1/6

So Adam has eaten 1/6 of the pizza.

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Simplifying the fraction by dividing both the numerator and denominator by 7: 7/42 = (1 × 7)/(6 × 7) = 1/6Hence, Adam has eaten 1/6 or 7/42 of the pizza.

Let's begin with the solution by calculating the fraction of the pizza that has been consumed:

Pizza's central angle = 360°

The central angle of Adam’s first piece = 35°

The central angle of Adam’s second piece = 25°

The total central angle of Adam's pizza pieces = 35° + 25° = 60°

The fraction of pizza was eaten by Adam = (Total central angle of Adam's pizza pieces)/(Central angle of one whole pizza)Fraction of pizza eaten by Adam = 60/360 = 1/6So,

Adam has eaten 1/6 of the pizza. Now, we can represent 1/6 as a fraction in which the numerator and denominator have the same value.

We do this by multiplying the numerator and denominator of the fraction by 7/7.

Thus, we get:1/6 = (1 × 7)/(6 × 7) = 7/42Therefore,

Adam has eaten 7/42 of the pizza.

Simplifying the fraction by dividing both the numerator and denominator by 7:7/42 = (1 × 7)/(6 × 7) = 1/6Hence, Adam has eaten 1/6 or 7/42 of the pizza.

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