A teacher takes 5 children on a field trip to a geological museum. As a souvenir, the group receives a bag of stones. When the teacher divides the stones evenly among the children, each child receives exactly 24. If 6 children had been present, how many stones would each child have received?

Using polynomial expansion, the resulting expression contains exactly 1001 terms that include all four variables a, b, c, and d, each to some positive power is 14.

According to the question,

For some particular value of N, when (a + b + c+ d+ 1)²  is expanded and like terms are combined, the resulting expression contains exactly 1001 terms that include all four variables a, b, c, and d, each to some positive power.

Polynomial expansion :

(a + b)ⁿ = Cₙ⁰ aⁿ + Cₙⁿ⁻¹ aⁿ⁻¹ b + ....+ Cₙⁿbⁿ

\(x_{a} + x_{b} + x_{c} + x_{d} + x_{1}\) =N

\(x_{a} > 1 x_{b} > 1x_{c} > 1x_{d} > 1x_{1}\) >0

Positive integer solution:

\(x_{a} + x_{b} + x_{c} + x_{d} + x_{1}\) = N+1

Non - Negative integer solution:

\(x_{a} + x_{b} + x_{c} + x_{d} + x_{1}\) =N -4

C⁴ₙ = 1001

1001 = 7.11.13 after some trial and error we find  that is N is equal to 14.

Hence, using polynomial expansion, the resulting expression contains exactly 1001 terms that include all four variables a, b, c, and d, each to some positive power is 14.

Learn more about Positive Power:

https://brainly.com/question/14786498

#SPJ4

The area of the prism is given by the sum of the areas of all the parts that compose the prism.

The parts that compose the prism are given as follows:

Hence the area of the prism is given as follows:

A = 3 x 5 + 5 x 10 + 2 x 1/2 x 3 x 10

A = 95 m².

The volume is given by the multiplication of the base area by the height, hence:

Thus:

V = 15 x 10 = 150 m³.

More can be learned about the area and the volume of a prism at https://brainly.com/question/463363

#SPJ1

The volume of the cube after removing a sphere of diameter 10cm from it = 476.67 cm³

Volume of a cube = \(side^{3}\) (side = side of the cube)

What is the volume of a sphere?

Volume of a sphere = \(\frac{4 \pi (radius)^{2} }{3}\) (radius = radius of the sphere)

Given:

Total volume of the cube = \(10^{3}\) = 1000 cm³

Volume of the sphere = \(\frac{4\pi (5)^{3} }{3}\) = 523.33 cm³

Volume of the cube after removing the sphere of diameter 10cm = Total volume of the cube - Volume of the sphere = 1000 - 523.33 = 476.67 cm³

Hence, the required volume of the cube = 476.67 cm³

To learn more about the volume of shapes, refer to the link: https://brainly.com/question/16717260

#SPJ4

Answer:

35.2 ; 4.75

Step-by-step explanation:

Given that :

Mean age, μ = 35.2 years

Standard deviation, σ = 9.5 years

The mean of the sampling distribution will be equal to the population mean, μ, this is according to the central limit theorem.

The standard deviation of the sampling distribution, is given by : σ/√n

n = sample size = 4

SD of sampling distribution = 9.5/√4 = 9.5 / 2 = 4.75

Answer:

The answer is 12

Step-by-step explanation:

If there are 36 gallons per 3 min. To find the per min you would divide 36 by 3.

Using Conditional probability,

The probability that after a fair coin is tossed 5 times that the result will contain 3 'heads' given that the first toss was a 'head' is 3/5.

A fair coin is tossed five times .

so, total possible outcomes = 2⁵ = 32

Let A: head observed on first toss

B: 3 heads observed.

Probability that the first toss was head (p) = 1/2

we have to find out probability that on 5 tosses result contain 3 heads given that first toss wss head .

Using Binomial Probability distribution

P (X= x) = ⁿCₓ (p)ˣ (q)⁽ⁿ⁻ˣ⁾

x = 3 , n = 5

we get , P(B) = P(X= 3 ) = ⁵C₃(1/2)³(1/2)²

=> P(B) = 10 ×(1/2)⁵= 5/16

since the first toss is fixed as a head.

P(A ∩ B) =P( X₁ = 1 and X₂+ X₃ + X₄ + X₅) = P(X₁= 1)P(X₂+X₃+X₄+X₅)

= 1/2 × ⁴C₂ (1/2)⁵ = 3 (1/2)⁴

The required probability is , Conditional probability P(B/A) = P(A ∩ B)/P(A)

= 3(1/2)⁴/ 5/16 = 3/5

Hence , the required probability is 3/5.

To learn more about Conditional Probability, refer:

https://brainly.com/question/10567654

#SPJ4

Answer:

In a rhombus, opposite angles are equal, therefore:

m∠S = m∠U = a + 62 degrees

And

m∠R = 2a - 80 degrees

The opposite angles of a rhombus are supplementary, meaning they add up to 180 degrees. So,

m∠S + m∠R = 180 degrees

Therefore,

a + 62 + 2a - 80 = 180

Solving for a, we get:

a = 59

So,

m∠S = a + 62 = 59 + 62 = 121 degrees

The coefficient of correlation (r) being -0.30 indicates a weak negative linear relationship between the variables x and y. The value of r ranges from -1 to 1, where -1 indicates a perfect negative linear relationship, 0 indicates no linear relationship, and 1 indicates a perfect positive linear relationship. Therefore, an r value of -0.30 suggests that there is a weak negative relationship between x and y, but the relationship is not strong enough to be considered a significant predictor of y based on x.

The negative value of r indicates that as the value of x increases, the value of y tends to decrease, although the relationship is weak. It is important to note that while a weak correlation does not necessarily imply causation, it does suggest that there may be some underlying relationship between the variables that should be further explored. Therefore, it is recommended to use other statistical measures in conjunction with the coefficient of correlation to determine the strength and significance of the relationship between x and y.

To learn more about coefficient of correlation : brainly.com/question/29704223

#SPJ11

There are MULTIPLE answers to this question.

- How much more snow is added to the lawn every minute

- The change in the amount of snow on the lawn for every one additional minute

- The rate of change in the amount of snow per minute

Hope this helps!

0.015 represents how much snow is added to the lawn every minute

An equation is a mathematical statement which equate two algebraic expressions.

So to make 0.015x represent how many inches of snow were on the lawn, 0.015 should represent the amount of snow is added to the lawn per minute.

So, 0.015 represents how much snow is added to the lawn every minute

Find out more about 'Linear Equations' here:  https://brainly.com/question/2226590

#SPJ2

Answer:

\(6\sqrt{2}\)

Step-by-step explanation:

let the diagonal be d

using the pythagoras theorem,

\(6^{2} + 6^{2} = d^{2}\)

\(d=\sqrt{72}\)

which in radical form is

\(6\sqrt{2}\)

The coordinates of the point N dividing the line LM, with L(-3, 1) and N(4, 9) in the ratio 2:3, found using the internal section formula is \(\left(-\frac{1}{5} ,\, \frac{21}{5} \right)\)

A ratio indicates the number of times one quantity a is contained in another quantity b.

The coordinates of the point L = (-3, 1), the coordinates of the point M = (4, 9)

The ratio in which the point N divides LM = 2:3

The specified ratio of the sides are;

\(\dfrac{LN}{MN} = \dfrac{2}{3}\)

The coordinates of the points N is found using the internal section  formula as follows;

\(C(x, y) = \left(\frac{m\times x_2 + n\times x_1}{m+n} , \,\frac{m\times y_2 + n\times y_1}{m+n} \right)\)

Where;

(x₁, y₂), and (x₂, y₂) are the endpoints of the line

m:n is the ratio the point C divides the line

Which indicates;

\(N(x, y) = \left(\frac{2\times 4 + 3\times (-3)}{2+3} , \,\frac{2\times 9 + 3\times 1}{2+3} \right) = \left(-\frac{1}{5} ,\, \frac{21}{5} \right)\)

Learn more about the internal section  formula here:

https://brainly.com/question/17218155

#SPJ1

The uniform continuous probability for the interval (25 < x < 45) within the uniform distribution U(15, 65) can be found by calculating the proportion of the total range that falls within that interval.

To calculate the probability, we need to determine the length of the interval (45 - 25) and divide it by the length of the entire range (65 - 15).

Length of the interval: 45 - 25 = 20

Length of the entire range: 65 - 15 = 50

Now, we divide the length of the interval by the length of the entire range to obtain the probability:

Probability = (Length of interval) / (Length of entire range) = 20 / 50 = 0.4

Therefore, the uniform continuous probability for p(25 < x < 45) within the uniform distribution U(15, 65) is 0.4, rounded to one decimal place.

To learn more about uniform continuous probability click here brainly.com/question/13257446

#SPJ11

Answer:

11/31 11/12 11/5

Step-by-step explanation:

because 11/5 is a mixed number and the bigger the denominator the smaller the number

Answer:

5

Step-by-step explanation:

The solutions to the equation -2p² - 5p + 1 = 7p² + p are p = (1 + √2) / (-3) and p = (1 - √2) / (-3).

To solve the equation -2p² - 5p + 1 = 7p² + p using the quadratic formula, we first rearrange the equation to bring all terms to one side:

-2p² - 5p + 1 - 7p² - p = 0

Combining like terms, we get:

-9p² - 6p + 1 = 0

Now, we can apply the quadratic formula, which states that for an equation of the form ax² + bx + c = 0, the solutions are given by:

p = (-b ± √(b² - 4ac)) / (2a)

In our case, a = -9, b = -6, and c = 1. Plugging these values into the quadratic formula, we have:

p = (-(-6) ± √((-6)² - 4(-9)(1))) / (2(-9))

Simplifying further:

p = (6 ± √(36 + 36)) / (-18)

p = (6 ± √72) / (-18)

p = (6 ± 6√2) / (-18)

Factoring out a common factor of 6:

p = (6(1 ± √2)) / (-18)

Simplifying the fraction:

p = (1 ± √2) / (-3)

For more questions on quadratic formula

https://brainly.com/question/1214333

#SPJ8

The measure of each base angle of the isosceles triangle is given as follows:

80º.

An isosceles triangle is an angle that has two equal side lengths, meaning that the base angles also have the same measure.

For the top angle of the isosceles triangle ABC, we have that the semicircle, which has an arc length of 180º, is divided into 9 regions, hence it's measure is calculated as follows:

Top angle = 180º/9 = 20º.

For the base angles, each of them have the same measure, which we are going to call x.

The sum of the measures of the internal angles is of 180º, hence the value of x can be found as follows:

20 + 2x = 180

2x = 160

x = 160/2

x = 80º.

More can be learned about angle measures at https://brainly.com/question/25716982

#SPJ1

Answer:

sorry i think a

Step-by-step explanation:

Answer:

x = 5

Step-by-step explanation:

14x + 1 + 14x + 1 + 38 = 180 , that is

28x + 40 = 180 ( subtract 40 from both sides )

28x = 140 ( divide both sides by 28 )

x = 5

Answer:

\({ \tt{ \tan(x) = \frac{opposite}{adjacent} }} \\ \\ { \tt{ \tan( \theta) = \frac{30}{16} }}\)

Subtraction is a mathematical operation that reflects the removal of things from a collection. The amount by which Rico weighs more than Biscuit is 23.6 kilograms.

Subtraction is a mathematical operation that reflects the removal of things from a collection. The negative symbol represents subtraction.

Given Rico weighed 45.7 kilograms, Mocha weighed 15.5 kilograms, and Biscuit weighed 22.1 kilograms. Therefore, the amount by which Rico weighs more than Biscuit is,

Difference = 45.7 kgs - 22.1 kgs = 23.6 kgs

Hence, the amount by which Rico weighs more than Biscuit is 23.6 kilograms.

Learn more about Subtraction:

https://brainly.com/question/1927340

#SPJ1

Answer:

m= 18g

Step-by-step explanation:

m = 18 × g

Answer: $1.05 per orange

Step-by-step explanation:

12.33 - 2.88 = 9.45

9.45 ÷ 9 = 1.05

(a) Is this an observational study or an experiment?

ans=Observational study – no treatment was imposed

(b) What are the explanatory and response variables?

ans=explanatory – time spent in child care from birth to age 4 response – adult rating of children’s behavior

d) Does this study show that child care causes children to be more aggressive? Explain.

No. Observational studies cannot show cause-and-effect

Researchers=someone whose job is to study a subject carefully, especially in order to discover new information or understand the subject better: She is a leading researcher in the field.

learn more about researcher here

https://brainly.com/question/25257437

#SPJ4

Answer:

A) -20

Step-by-step explanation:

\(4y(4y-4)-4(-4y+5+4y^2)\\\\16y^2-16y + 16y-20-16y^2\\\\-20\\\)

The required equation function j which is the transformation of \(f(x)= x^{1/2}\) is  \(j(x)= (x+2)^{1/2}\) .

As we can see in the graph the functions j and f are similar but the graph of j is 2 units left of f. So, function f(x) has a domain of real number greater than equal to zero, while as the function j is 2 units left of f then its equation is given as  \(j(x)= (x+2)^{1/2}\)  where the domain of j is all real numbers greater then or equal to -2.

Thus, the equation of function j is the transformation of \(f(x)= x^{1/2}\) is  \(j(x)= (x+2)^{1/2}\) .

Learn more about the transformation of the function here:

https://brainly.com/question/26896273

#SPJ1

In the question the graph is missing, a graph has been added on behalf of the incomplete question.

we can conclude that if the game is played 8 times, the probability of winning X prizes is given by the binomial probability distribution and the probability distribution for X is 0.43, 0.39, 0.15, 0.03, 0, 0, 0, 0, 0. If the game is played 50 times, then the expected number of prizes won is 5.

a) Probability distribution of the number of prizes won in 8 games is given by the binomial probability distribution.

As the probability of winning a prize in one game is 0.1, probability of not winning a prize is 0.9.

If X is the number of prizes won in 8 games, then the probability of winning X prizes is given by the formula:

P(X = x)

= nC x * p ˣ* (1-p)ᵃ     (a=n-x),

where n = 8, p = 0.1 and x varies from 0 to 8.

The probability distribution for X is as follows:

X     0       1       2       3       4       5       6       7       8

P(X) 0.43 0.39 0.15 0.03 0.00 0.00 0.00 0.00 0.00

b) If the game will be played 50 times, then the expected number of prizes won is given by the formula:

E(X) = n*p

= 50*0.1

= 5.

Therefore, we can expect 5 prizes to be won if the game is played 50 times.

Hence, we can conclude that if the game is played 8 times, the probability of winning X prizes is given by the binomial probability distribution and the probability distribution for X is 0.43, 0.39, 0.15, 0.03, 0, 0, 0, 0, 0. If the game is played 50 times, then the expected number of prizes won is 5.

To know more about probability visit:

brainly.com/question/31828911

#SPJ11

Answer:

x=11

Step-by-step explanation:

65+3x+4+78=180

Combine like terms

147+3x=180

Subtract 147 on both sides

3x=33

Divide by 3

x= 11

Answer: 11 degrees

Step-by-step explanation:

65+3x+4+78=180

147+3x=180

3x=33

x=11