At the Pineville County Fair popcorn stand, Vivian scoops popcorn from a large container to fill smaller bags to sell. As she fills the bags, the amount of popcorn in the container decreases. This situation can be modeled as a linear relationship.

Answer:

$20

Step-by-step explanation:

25% off of 100% = 75%
0.75x=15, where x is the original bill

x=20

(a) The total number of ways to distribute the apples and pears subject to the conditions is = 135,124 ways.

(b) The total number of ways to distribute the apples and pears subject to the conditions is: 3 \times 10 \times [tex]3^7[/tex] = 34,650 ways.

Combinatorics is a branch of mathematics that deals with counting and arranging the possible outcomes of different arrangements and selections of objects.

a) To distribute seven identical apples and six identical pears to three distinct people such that each person has at least one pear, we can use the principle of inclusion-exclusion. Let's denote the three people as A, B, and C.

First, we can distribute the six pears in[tex]{6+3-1 \choose 6} = {8 \choose 6} = 28[/tex] ways using stars and bars.

Next, we can distribute the seven apples to the three people without any restrictions, which can be done in [tex]3^7[/tex] ways.

However, this overcounts the cases where one or more people receive no pears. There are [tex]{3 \choose 1}[/tex] ways to choose one person who does not receive a pear, and then [tex]2^7[/tex] ways to distribute the apples among the remaining two people.

Similarly, this also overcounts the cases where two people receive no pears. There are [tex]{3 \choose 2}[/tex] ways to choose two people who do not receive a pear, and then [tex]1^7[/tex] way to distribute the apples among the remaining person.

Finally, we need to add back the cases where all three people receive no pears, which is just one way (each person receives no pears).

Thus, the total number of ways to distribute the apples and pears subject to the conditions is:

[tex]28 \times - {3 \choose 1} \times + {3 \choose 2} \times - {3 \choose 3} \times = 135,124 ways.[/tex]

b) To distribute seven distinct apples and six distinct pears to three distinct people such that each person has at least one pear, we can first choose the person who receives the remaining pear in [tex]{3 \choose 1} = 3[/tex] ways. Then we can distribute the pears in [tex]{6-1 \choose 3-1} = {5 \choose 2} = 10[/tex] ways, leaving one pear for the chosen person and distributing the other two pears among the remaining two people using stars and bars.

Next, we can distribute the seven distinct apples to the three people without any restrictions, which can be done in [tex]3^7[/tex] ways.

Thus, the total number of ways to distribute the apples and pears subject to the conditions is:

[tex]3 \times 10 \times 3^7[/tex] = 34,650 ways.

To learn more about combinatorics from the given link:

https://brainly.com/question/31476844

#SPJ1

The graph is attached, from the graph the zeros are

x = -7, 1 and 3

The polynomial in factored form. P(x) = (x + 7) (x - 1) (x - 3)

The zeros of the polynomial function given as P(x) = x³ + 3x² − 25x + 21 is solved using graphical method

From the graph are deduced to be

x = -7, x = 1 and x = 3

From the zeros the factored form of the equation is written as

x = -7, x + 7 = 0

x = 1, x - 1 = 0

x = 3, x - 3 = 0

hence factored form of the polynomial is, P(x) = (x + 7) (x - 1) (x - 3)

Learn more about factored form of polynomial at

https://brainly.com/question/1358189

#SPJ4

The probability that the mean of a sample of 55 families will be between 17 and 18 pounds is approximately 71.55%.

σ_sample = σ / sqrt(n)

σ_sample = 2.5 / sqrt(55)

σ_sample ≈ 0.336

Now, we can standardize the values of 17 and 18 pounds using the Z-score formula. This will tell us how many standard deviations away from the population mean these values are:

Z = (X - μ) / σ_sample

Z_17 = (17 - 17.2) / 0.336 ≈ -0.595

Z_18 = (18 - 17.2) / 0.336 ≈ 2.381

Now we will use the standard normal distribution (Z-distribution) to find the probabilities corresponding to these Z-scores. You can use a Z-table, statistical software, or an online calculator to find these probabilities.

P(Z < -0.595) = 0.2760

P(Z < 2.381) = 0.9915

To find the probability that the mean of a sample of 55 families will be between 17 and 18 pounds, we will find the difference between the probabilities corresponding to the two Z-scores:

P(17 < X < 18) = P(Z < 2.381) - P(Z < -0.595)

P(17 < X < 18) = 0.9915 - 0.2760

P(17 < X < 18) ≈ 0.7155

So, the probability that the mean of a sample of 55 families will be between 17 and 18 pounds is approximately 71.55%.

Read more on probability here:https://brainly.com/question/24756209

#SPJ4

The value of g(-5) is 7.5, g(-2) is 4, and g(5) is 7.5 if the value of function g(x) = 1/2x²-5 if x≠-2 and g(x) = 4 if x = -2.

A function is a rule that associates each element in a set (called the domain) with a unique element in another set (called the range). A function takes an input from the domain and produces an output in the range.

To find the value of g(-5), substitute the value in given function

g(-5) = 1/2×(-5)² - 5

= 1/2×25 - 5

= 12.5 - 5 = 7.5

The value of g(-2) is 4 as it is defined in the question only.

To find the value of g(5), substitute the value in given function

g(5) = 1/2×(5)² - 5

= 1/2×25 - 5

= 12.5 - 5 = 7.5

To know more about function here

https://brainly.com/question/12431044

#SPJ1

Statements (iii) and (iv) are not equivalent, as they also have different truth values for some combinations of p and q.

What are proportions?

In mathematics, a proportion is a statement that two ratios are equal. It expresses the relationship between two or more quantities that are directly proportional to each other. A proportion can be represented as an equation of the form:

a/b = c/d

To construct the truth tables, we assign a truth value of either true or false to each statement. We use "T" to represent true and "F" to represent false. Then, we evaluate the truth value of each statement for all possible combinations of truth values for the variables involved. In this case, there is only one variable, which is whether or not someone knows CPR.

Let's use the variable p to represent the statement "someone knows CPR" and q to represent the statement "someone is a paramedic".

Then, the statements can be rewritten as follows:

i. p is necessary for q.

ii. p is sufficient for q.

iii. not p is necessary for not q.

iv. not p is sufficient for not q.

The truth tables for each statement are as follows:

i.

p q p is necessary for q

T T T

T F F

F T T

F F T

ii.

p q p is sufficient for q

T T T

T F F

F T T

F F T

iii.

p q not p is necessary for not q

T T F

T F T

F T F

F F T

iv.

p q not p is sufficient for not q

T T F

T F T

F T F

F F T

From the truth tables, we can see that statements (i) and (ii) are not equivalent, as they have different truth values for some combinations of p and q. Similarly, statements (iii) and (iv) are not equivalent, as they also have different truth values for some combinations of p and q.

To learn more about proportions from given link:

brainly.com/question/26617614

#SPJ1

Algebraic reasoning involves using mathematical structures and processes to analyze and solve problems, and it is an important part of mathematical thinking and problem solving.

What is Algebraic expressions.?

An algebraic expression is a combination of variables, constants, and mathematical operations such as addition, subtraction, multiplication, division, and exponentiation. Algebraic expressions can contain one or more variables, which are usually represented by letters, and they can be simplified or evaluated by substituting numerical values for the variables.

For example, the expression "3x + 2y - 5" is an algebraic expression that contains two variables, x and y, and three constants, 3, 2, and 5. The expression can be evaluated or simplified by substituting specific values for the variables. For instance, if we let x = 2 and y = 1, then the expression becomes "3(2) + 2(1) - 5", which simplifies to 3.

Skip counting, clapping hands rhythmically, and calling out numbers in unison can be helpful activities for students to develop and practice early mathematical concepts. However, they are not necessarily examples of algebraic reasoning.

Algebraic reasoning involves using mathematical symbols and equations to represent and solve problems. It typically involves working with variables, manipulating expressions, and solving equations.

Some examples of algebraic reasoning for students might include:

Writing and solving simple equations, such as "2x + 3 = 7"

Using variables to represent unknown quantities in problems, such as "If x is the number of apples John has, and he gives away 3, how many apples does he have left?"

Recognizing and extending patterns, such as identifying the rule for a sequence of numbers and predicting the next term in the sequence

Simplifying expressions and using the distributive property, such as simplifying "3(x+2) + 2(x+1)" to "5x + 8"

Overall, algebraic reasoning involves using mathematical structures and processes to analyze and solve problems, and it is an important part of mathematical thinking and problem solving.

To know more about Algebraic expressions. visit;

https://brainly.com/question/4344214

#SPJ4

The equation of this circle in standard form include the following: D. (x + 7)² + (y - 10)² = 11².

In Mathematics and Geometry, the standard form of the equation of a circle is represented by the following mathematical equation;

(x - h)² + (y - k)² = r²

Where:

Based on the information provided, we have the following parameters:

By substituting the given parameters into the equation of a circle formula, we have the following;

(x - h)² + (y - k)² = r²

(x - (-7))² + (y - 10)² = 11²

(x + 7)² + (y - 10)² = 11²

Read more on equation of a circle here: brainly.com/question/15626679

#SPJ1

Note that the answers to the above transformation prompt is above as follows:

1. Angle PQR

2. Segment PS

3. Scale factor =  3/2 = 1.5

1. Angle PQR in the scaled copy corresponds to angle ABC in polygon ABCD in polygon PQRS. In the two separate figures, they are both in the same position. They complement each other.

2. In the scaled copy, segment PS is the section in polygon PQRS that corresponds to segment AD in polygon ABCD. They are similar in appearance and appear in the same location.

3. The scale factor is the ratio of any of the segment lengths.

Let's consider segments lengths of AD in ABCD, which corresponds to segment PS in PQRS.

AD = 2 units

PQ = 3 units

Scale factor from ABCD to PQRS = PQ/AD = 3/2 = 1.5

The whole number tells us that ABCD is scaled up or enlarged to given a bigger polygon, PQRS

Learn more about Scale factors:
https://brainly.com/question/30215044
#SPJ4

The equation represents the proportional relationship between the number of bag and cost of cereal bag is y = 7.25x

Number of bags = 3

Cost of three bags of cereal = 21.75

Let the number of bags of cereal bought = x

The cost paid for the bag = y

They both are directly proportional

3/x = 21.75/y

3y = 21.75x

on dividing both side with 3 we will get cost of one bag of cereals

3y/3 = 21.75x/3

y = 7.25x

The proportional relationship between a number of bags of cereal and the cost of cereal is y = 7.25

To know more about  proportional click here :

https://brainly.com/question/30657439

#SPJ4

The calculated length of the segment EF is (c) 5

from the question, we have the following parameters that can be used in our computation:

The trapezoid

The length of segment EF can be calculated using

EF = 1/2 * Sum of BC and AD

using the above as a guide, we have the following:

EF = 1/2 * (3.3 + 6.7)

Evaluate

EF = 5

Hence, the length of segment EF is (c) 5

Read more about midsegments at

https://brainly.com/question/7423948

#SPJ1

The mean change in heart rate is 0.28 , around to 2 decimal places.

What is mean?

In statistics, the mean is nothing but  the measures of central tendency, apart from the mode and median. Mean is the average of the given set of values or data. It denotes the equivalent distribution of values for a given data set. To calculate the mean of given data, we need to add the total values given in a datasheet and  then divide the sum by the total number of values.

The change in heart rate for the 10 individuals are as follows

(1.8, -0.4, -0.6, 0.4, 0.2, -1.0, 1.6, 0.8, 2.0, -2.0).

So the data is given by,

(1.8, -0.4, -0.6, 0.4, 0.2, -1.0, 1.6, 0.8, 2.0, -2.0).

Mean of the given data = sum of all data/ number of data

Mean = (1.8-0.4 -0.6+0.4+ 0.2 -1.0+1.6+ 0.8+2.0-2.0)/10

         = 2.8/10

         =0.28

Hence, the mean change in heart rate is 0.28 , around to 2 decimal places.

To know more about mean

https://brainly.com/question/1136789

#SPJ4

Answer:

[tex]m = \frac{ - 1 - 2}{3 - 0} = \frac{ - 3}{3} = - 1[/tex]

We know that b, the y-intercept, is 2, so:

[tex]y = - x + 2[/tex]

The calculated value of the product of 0.8 and 0.27 is 0.216

From the question, we have the following parameters that can be used in our computation:

What is the product of 0.8 and 0.27?

When represented as a product expression, we have

0.8 * 0.27

Evaluating the products

so, we have the following representation

0.8 * 0.27 = 0.216

This means that the product of 0.8 and 0.27 is 0.216

Read mroe about expressions at

https://brainly.com/question/4541471

#SPJ1

Step-by-step explanation:

a)

with step 1 to 3 we see that the numbers of squares serving the length of the sidelines is following the step number 1:1.

so, step 1 has a side length of 1 square.

step 2 has a side length of 2 squares.

step 3 has a side length of 3 squares.

so, we expect step 4 to have a side length of 4 squares.

but the first statement says "5-by-5", so, the side length is 5.

therefore, this statement is false.

and yes, we expect step 15 to have a side length of 15.

and the shift of the single square from the bottom left to the top right is also correct.

therefore, the statement is true.

b)

step 4 = 4×4 = 16 squares

step 15 = 15×15 = 225 squares

c)

no.

just look at step 1.

y = 1² - 1 = 1 - 1 = 0.

but I can clearly see that there is 1 square.

d)

it makes an n×n array. which is the same as n².

By solving differential equation for [tex]du/dt = e^5u + 3t.[/tex] The u =[tex](-1/5) ln[25t + e^{-5u} - 26].[/tex]

We have to separate variables first:

[tex]du/e^5u = dt + 3t/e^{5} u du[/tex]

Now, we will integrate sides to their variables:

∫ du/[tex]e^{5u}[/tex]= ∫ (dt + 3t/[tex]e^{5u}[/tex]) du

Using substitution:

let w = 5u, then du = dw/5:

1/5 ∫ dw = [tex]e^{-w}[/tex]∫ (dt + 3t/[tex]e^w[/tex] (du/5)

Integrating both sides:

-1/5 [tex]e^{-w}[/tex]= t + (1/25) [tex]e^{-w}[/tex] + C

Substituting back w = 5u:

-1/5 [tex]e^{-5u}[/tex]= t + (1/25) [tex]e^{-5u}[/tex] + C

Using initial condition, u(0) = 5:

-1/5 [tex]e^{-25}[/tex] = 0 + (1/25) [tex]e^{-25}[/tex] + C

C = -26/25

The solution to differential equation with initial condition is:

-1/5 [tex]e^{-5u}[/tex] = t + (1/25) [tex]e^{-5u}[/tex]- 26/25

When we solve for u, we have:

u = (-1/5) ln[25t + [tex]e^{-5u}[/tex]- 26].

Read more about Differential Equations

brainly.com/question/1164377

#SPJ4

Sarah's score of 85 puts her at the 40th percentile in her class.

This means that 40% of the students in her class got a score equal to or below her score, and 60% of the students got a score above her score.

Percentile is a statistical measure that indicates the percentage of a population that is below or equal to a certain score or value.

If a student scores in the 90th percentile, it means that they have performed better than 90% of the other students in their class.

To calculate the percentile rank of a student, you need to follow these steps:

Determine the total number of scores in the class, including the student's score.

Sort the scores in ascending order.

Count the number of scores that are below the student's score.

This is called the student's rank.

Divide the student's rank by the total number of scores and multiply the result by 100.

This gives you the student's percentile rank.

Let's say that a class of 30 students took a math test, and one student, Sarah, received a score of 85.

To determine Sarah's percentile rank, you would follow these steps:

The total number of scores in the class is 30.

Sorting the scores in ascending order, Sarah's score of 85 falls between the 21st and 22nd scores.

There are 20 scores below Sarah's score of 85.

Her rank is 21.

Dividing Sarah's rank (21) by the total number of scores (30) and multiplying by 100 gives us her percentile rank:

(21/30) × 100 = 70.

This means that Sarah scored higher than 70% of the other students in the class.

Percentile rank is a useful measure for comparing individual scores to a larger group or population.

Percentile rank, you can easily determine a student's score falls in comparison to their peers.

For similar questions on Percentile

https://brainly.com/question/2263719

#SPJ11

The expression equals to the fraction of squared that is shaded blue is 3/8

Fraction is defined as the parts of a whole the number is expressed as a quotient, in which the numerator is divided by the denominator

The whole part here is the total number of boxes and its blue shaded part is defined as the part of it.

The Numerator represents the shaded part = 3

The Denominator represent the whole squared boxes = 8

The total number of square boxes is 8

the number of blue shaded boxes is 3

Fraction of the square = 3/8

To know more about fraction click here :

https://brainly.com/question/10354322

#SPJ4

The question is incomplete the complete question is :

Which expression equals the fraction of the square that is shaded blue?

ItI > teritical I.p value <0.05, we fail to reject num hypothesis

Test statistic , t= x1-x2/s1.s1/n1+s2.s2/n2

=26-32/3.3/18+2.2/24

=-6/0.815=-7.3484

iti=7.2484

Teritical at >=0.05 I df=n1+n2+n3=40

Static testing is the process of reviewing the code and creating documents and requirements before it is performed to detect errors instead of actually running the code. The major objective is to identify faults early in the development process because it is typically simpler to identify the potential causes of failures in this way.

A sort of testing called static testing is done on a piece of software without running the actual code. We examine and validate the product and its supplementary materials during testing.

Dynamic testing, in contrast, is a sort of testing done on software while the code is being executed.A test statistic is a figure obtained from a statistical analysis. It explains how far your observed data is from the null hypothesis, which states that there is no correlation between the variables or distinction between the sample groups.

To learn more about Teritical , visit

https://brainly.com/question/29850660

#SPJ4

Volume of "region" which lies between the paraboloid z = 24 - x² - y² and the cone z = 2√x² + y² is 536.2 cubic units.

In order to find the volume, we first convert in the polar-coordinate;

We get,

Cone : z = 2√x² + y² ⇒ 2r;  and

Paraboloid : z = 24 - x² - y² ⇒ 24 - r²;

So, the point of intersection can be found by;

2r = 24 - r²,

r² - 24r + 2r = 0,

r = 4, -6.

Since, r≥0, we have r=4;

The intervals are : 0≤r≤4 and 0≤θ≤2π;

So, the Volume is written as : [tex]\int\limits^{2\pi}_0 \int\limits^4_0[/tex] ((24-r²) - 2r)r.dr.dθ

⇒  [tex]\int\limits^{2\pi}_0 \int\limits^4_0[/tex] (24 - 2r - r²)r.dr.dθ

⇒  [tex]\int\limits^{2\pi}_0 \int\limits^4_0[/tex] (24r - 2r² - r³)dr.dθ

⇒  [tex]\int\limits^{2\pi}_0[/tex] [12r² - 2r³/3 - r⁴/4]⁴₀,

⇒  2π [192 - 128/3 - 64] = 512π/3 ≈ 536.2 cubic units.

Therefore, the volume of the region is 536.2 cubic units.

Learn more about Volume here

https://brainly.com/question/30785714

#SPJ4

The given question is incomplete, the complete question is

Find the volume of the region that lies between the paraboloid z = 24 - x² - y² and the cone z = 2√x² + y².

To divide exponents with different bases and powers, you need to first factor out the common factors from the exponents. This will help you simplify the expression and make it easier to solve.

For example, let's say you have the expression 4x^3 / 2x^2. To divide these exponents, you need to factor out the common factors. In this case, the common factor is x^2, so you can simplify the expression as follows:

4x^3 / 2x^2 = (4/2) * (x^3/x^2) = 2x^(3-2) = 2x

So the answer is 2x.

In general, to divide exponents with different bases and powers, you need to:

1. Factor out the common factors from the exponents
2. Divide the coefficients (if there are any)
3. Subtract the exponents of the common factors

By following these steps, you can easily divide exponents with different bases and powers.
To divide exponents with different bases and powers, follow these steps:

1. Convert the bases to their prime factorization (if needed).
2. Divide the exponents with the same base by subtracting the exponent in the denominator from the exponent in the numerator.
3. Combine the results back together as a simplified expression.

Remember, you can only directly divide exponents with the same base. If bases are different, you'll need to factorize them to find common factors.

Visit here to learn more about exponents brainly.com/question/5497425
#SPJ11

The equation to complete the square for x² + 3x = 6 is 3/2 (option a)

To complete the square for the equation x² + 3x = 6, we need to add a specific number to both sides of the equation. The number we need to add is half of the coefficient of x, squared. In other words, we need to find (b/2)², where b is the coefficient of x.

In this equation, b is 3, so (b/2)² is (3/2)², which simplifies to 9/4. Therefore, to complete the square, we need to add 9/4 to both sides of the equation:

x² + 3x + 9/4 = 6 + 9/4

Now, we can rewrite the left side of the equation as a perfect square:

(x + 3/2)² = 33/4

Finally, we can solve for x by taking the square root of both sides of the equation:

x + 3/2 = ± √(33/4)

x = -3/2 ± √(33/4)

Therefore, the answer to the question is a) 3/2.

To know more about equation here

https://brainly.com/question/10413253

#SPJ4

A one-sided significance test provides a p-value of 0.4, which helps us make decisions about the null hypothesis in a statistical analysis. The p-value represents the probability of obtaining results as extreme or more extreme than the observed data, assuming the null hypothesis is true. In most cases, a significance level (alpha) of 0.05 is used as a threshold for determining statistical significance.

When we compare the p-value of 0.4 to the significance level of 0.05, we notice that the p-value is greater than the alpha value. This indicates that we do not have sufficient evidence to reject the null hypothesis. In other words, the results of the study are not statistically significant, and we cannot conclude that there is a significant effect or relationship between the variables of interest.

It is important to remember that failing to reject the null hypothesis does not necessarily mean that there is no effect or relationship between the variables; it merely suggests that the evidence is not strong enough to make such a claim. Further research or larger sample sizes might be needed to explore the relationship between the variables more accurately.

In conclusion, a one-sided significance test with a p-value of 0.4 suggests that we cannot reject the null hypothesis, as the results are not statistically significant at the commonly used alpha level of 0.05.

To know more about null hypothesis visit:

https://brainly.com/question/30821298

#SPJ11

For an integral value of [tex]I_1 = \int_{ -2}^{3} [2f(x) + 2] dx = 18 \\ [/tex] and

[tex]I_2 = \int_{ 1}^{-2} f(x)dx = -10 \\ [/tex], the computed value of integral [tex]\int_{ 1}^{3} f(x)dx[/tex] is equals to the -6. So, option(c) is right one.

In mathematics, an integral is the continuous process of a sum, which is used to calculate areas, volumes, and their properties. Integration is a way to sum up parts to the whole.

We have an integral say [tex]I_1 = \int_{ -2}^{3} [2f(x) + 2] dx = 18 \\ [/tex]

[tex]I_2 = \int_{ 1}^{-2} f(x)dx = 10 \\ [/tex]

We have to determine value of [tex]\int_{ 1}^{3} f(x)dx[/tex].

Using the properties of integral, consider integral [tex]I_1 = \int_{ -2}^{3} [2f(x) + 2] dx = 18\\ [/tex]

from distribution property, [tex]I_1 = \int_{ -2}^{3} 2f(x) dx + \int_{ -2}^{3} 2 dx = 18 \\ [/tex]

[tex]2 \int_{ -2}^{3} f(x) dx + [ 2x]_{ -2}^{3} = 18[/tex]

[tex]2 \int_{ -2}^{3} f(x) dx + 10 = 18[/tex]

[tex]2 \int_{ -2}^{3} f(x) dx = 8[/tex]

[tex]\int_{ -2}^{3} f(x) dx = 4[/tex]

Now, consider the required integral and rewrite, [tex]\int_{ 1}^{3} f(x)dx = \int_{ 1}^{-2} f(x)dx + \int_{ -2}^{3} f(x)dx \\ [/tex]

Substitute all known values of integrals

[tex]\int_{ 1}^{3} f(x)dx = 10 + 4 = 14 [/tex]

Hence, required value is 14.

For more information about integral, visit:

https://brainly.com/question/27419605

#SPJ4

The length of a wave does not directly determine its depth. The depth of a wave is influenced by various factors, including the shape and size of the seafloor,

The temperature and salinity of the water, and the strength and direction of the current. Therefore, without additional information about these factors, it is impossible to accurately determine the depth of a wave based solely on its length. I suggest seeking the advice of an expert in oceanography or physics for a more comprehensive explanation.

The length of a wave is the distance between two successive points in the same phase, such as from crest to crest or trough to trough. In your case, the wave length is given as 30 feet. However, without additional information about the wave, we cannot determine its depth solely based on its length.

Depth refers to the vertical distance from the surface to the lowest point of a wave (trough), and it's not directly related to the wave length. Therefore, we cannot accurately choose an option (a, b, c, or d) for the depth based on the provided information.

To know more about length click here

brainly.com/question/30625256

#SPJ11

The value of the missing side lengths such as b and d would be = 3.5 and 11.3

To calculate the side length of the given triangle, the sine rule must be obeyed. That is;

a/sinA= b/sin B

For the first triangle;

b = ?

B = 30°

a = 6

A = 60°

That is ;

b/ sin30° = 6/sin60°

Make b the subject of formula;

b = sin30×6/sin60

= 0.5×6/0.866025403

= 3.5

For second triangle:

a/sinA= d/sin D

where;

a = 8

A = 45°

d = ?

D = 90°

That is;

8/sin45° = d/sin90°

d =8×1/0.707106781

= 11.3

Learn more about triangles here:

https://brainly.com/question/28470545

#SPJ1

The outward flux is 81, and the counterclockwise circulation is 54.

To apply Green's theorem, we need to find the curl of the vector field:

curl(f) = (∂f_y/∂x - ∂f_x/∂y) = (8y - (-6x))i + ((-6x) - 8y)j = (8y + 6x)i - (8y + 6x)j = (8y + 6x)(i - j)

Now, we can use Green's theorem, which states that the counterclockwise circulation of a vector field around a closed curve C is equal to the outward flux of the curl of the vector field through the region enclosed by C. In this case, the curve C is a triangle bounded by y = 0, x = 3, and y = x.

The counterclockwise circulation of the vector field around C is:

∫_C f · dr = ∫_C (4[tex]y^{2}[/tex] - 3[tex]x^{2}[/tex])dx + (3[tex]x^{2}[/tex] - 4[tex]y^{2}[/tex])dy

We can break this into three line integrals, corresponding to the three sides of the triangle:

∫_L1 f · dr =    [tex]\int\limits^3_0 {(4y^{2}-3x^{2})} \, dx[/tex]   = 36

∫_L2 f · dr = [tex]\int\limits^3_0 {(3x^{2}-4x^{2})} \, dy[/tex] = -9

∫_L3 f · dr = [tex]\int\limits^3_0 {(4y^{2}-3y^{2})} \, dx[/tex] = 27

The total circulation is the sum of these three line integrals:

∫_C f · dr = 36 - 9 + 27 = 54

To find the outward flux of the curl of f through the region enclosed by C, we need to find the area of the triangle. The base of the triangle is 3, and the height is also 3, since y = x along the slanted side. Therefore, the area is (1/2)(3)(3) = 4.5.

The outward flux of the curl of f through the region enclosed by C is:

∫∫_R curl(f) · dA = ∫∫_R (8y + 6x)dA

where R is the region enclosed by C. We can integrate this over the triangular region R by breaking it into two integrals:

∫∫_R curl(f) · dA = ∫_0^3 ∫_0^x (8y + 6x)dydx + ∫_3^0 ∫_0^(3-x) (8y + 6x)dydx

= 81

As a result, the anticlockwise circulation is 54 and the outward flux is 81.

To learn more about flux here:

https://brainly.com/question/31745682

#SPJ4

Answer:

can you tell me the options please so I can answer it thank you

Alan would need a sample size of at least 447 to construct a 95% confidence interval for the proportion of those testing positive for COVID-19 who require hospitalization with a margin of error no more than 2%, using a prior estimate of 15%.

What is statistics?

Statistics is a branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of numerical data. It involves the use of methods and techniques to gather, summarize, and draw conclusions from data.

To find the sample size required for constructing a 95% confidence interval for the proportion of those testing positive for COVID-19 who require hospitalization with a margin of error no more than 2%, we can use the following formula:

n = [(z-score)² * p * (1-p)] / (margin of error)²

where:

n is the sample size

z-score is the critical value for a 95% confidence interval, which is 1.96

p is the prior estimate of the proportion, which is 0.15

margin of error is 0.02

Plugging in the values, we get:

n = [(1.96)² * 0.15 * (1-0.15)] / (0.02)²

n = 446.25

Rounding up to the nearest whole number, we get a required sample size of 447.

Therefore, Alan would need a sample size of at least 447 to construct a 95% confidence interval for the proportion of those testing positive for COVID-19 who require hospitalization with a margin of error no more than 2%, using a prior estimate of 15%.

To learn more about statistics from the given link:

https://brainly.com/question/28053564

#SPJ4

The length of the curve is [tex]e^ \pi[/tex] - 1 units.

To find the length of the curve [tex]x = e^t cos(t), y = e^tsin(t)[/tex], where 0 ≤ t ≤ π.

In this case, the curve is defined by the equations [tex]x = e^t cos(t)[/tex] and [tex]y = e^t sin(t)[/tex], where t is the variable that represents the parameter along the curve.

To find the length of this curve, we can use a formula known as the arc length formula.

L = ∫[a,b] √〖[dx/dt]〗² + 〖[dy/dt]² dt

where L represents the length of the curve, a and b are the starting and ending values of the parameter, dx/dt and dy/dt are the derivatives of x and y with respect to t.

To find the derivatives dx/dt and dy/dt. Using the product rule and chain rule of differentiation, we get:

[tex]dx/dt = e^tcos(t) - e^t sin(t) dy/dt = e^t sin(t) + e^t cos(t)[/tex]

Substituting these expressions into the arc length formula and simplifying, we get: L = ∫[0,π] [tex]e^tdt[/tex]

Integrating this expression with respect to t, we get: L = [tex]e^\pi[/tex]- 1

So the length of the curve is [tex]e^ \pi[/tex] - 1 units.

To know more about Integral here

https://brainly.com/question/18125359

#SPJ4