answer the following

Area under the curve y = 3x² + 1  which lies between y = 4 and y = 2 is

4 - [tex]\frac{8}{3}\sqrt{\frac{1}{3} }[/tex] sq, units.

Using microscopic slivers or stripes of the region, a definite integral is a formal estimate of the area beneath a function. Integrals can reflect a region's (signed) area, the total value of a function that changes over time, or the amount of a particular thing given its density.

By performing a definite integral between the two locations, one may determine the area under a curve between two points. Integrate y = f(x) between the limits of a and b to determine the area under the curve y = f(x) between x = a and x = b. Areas below the x-axis will have a negative result, while those above the x-axis will have a positive result.

The given function is y = 3x² + 1

when y = 2  value of x in the function is ± (√(1/3))

and, when y = 4  value of x in the function is ± (1)

Area under the curve y = 2 is

∫3x² + 1 . dx from x ± (√(1/3)).

= (x³ + x) from x ± (√(1/3)).= [tex]\frac{8}{3}\sqrt{\frac{1}{3} }[/tex]

Similarly for the curve y = 4

∫3x² + 1 . dx from x ±1

= (x³ + x) from x ± 1.= 4

Area under the curve y = 3x² + 1  which lies between y = 4 and y = 2 is

4 - [tex]\frac{8}{3}\sqrt{\frac{1}{3} }[/tex] sq, units.

To learn more about Definite integral refer to :

https://brainly.com/question/20464528

#SPJ1

The intercept and slope of the linear function are - 3 and  3 / 2. (Correct choice: C)

Herein we find the representation of a linear function, whose algebraic equation is shown below:

y = m · x + b

Where:

Please notice that intercept is found when x = 0 and slope can be determined by secant line formula:

m = Δy / Δx

Where:

First, determine the intercept of the linear function:

b = - 3

Second, calculate the slope of the line by secant line formula:

m = [3 - (- 6)] / [4 - (- 2)]

m = 9 / 6

m = 3 / 2

To learn more on linear functions: https://brainly.com/question/14695009

#SPJ1

Answer:

3.5 cups of sugar / 1.75 cups of sugar = 2

3.5 x 2 = 7 cups of flour

Kareem needs 7 cups of flour to make muffins.

The measure of tangent BC from the given figure is 12.6 units.

The Pythagoras theorem which is also referred to as the Pythagorean theorem explains the relationship between the three sides of a right-angled triangle. According to the Pythagoras theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides of a triangle.

Given that, Circle A has radius =5.8 and DC=8.1. BC is tangent line.

Here, AB=AD= 5.8 units and AC=AD+DC =5.8+8.1

= 13.9

Using Pythagoras theorem, we have

AC²=AB²+BC²

13.9²=5.8²+BC²

BC²= 193.21-33.64

BC²= 159.57

BC=12.6 units

Therefore, the measure of tangent BC is 12.6 units.

To learn more about the Pythagoras theorem visit:

brainly.com/question/21926466.

#SPJ1

There were originally 36 plants in the nursery.

From the question, we have that we can be able to show the total number of the plants as x.

Hence;

When he passes the  1st guard:

x - (0.5x+2)

x -0.5x - 2

0.5x-2 plants left

When he passes the 2nd guard

(0.5x-2) - (0.5(.5x-2)+2)

0.5x - 2 - (0.25x-1+2)

0.5x - 2 - (0.25x+1)

0.5x - 2 - 0.25x - 1

When he passes the 3rd guard

(0.25x-3) - (0.5(0.25x-3)+2) = 1

0.25x - 3 - (0.125x-1.5+2) = 1

0.25x - 3 -(0.125x+.5) = 1

0.25x - 3 - 0.125x - 0.5 = 1

0.125x - 3.5 = 1

0.125x = 1 + 3.5

Then;

x = 4.5/0.125

x = 36 plants

Learn more about plants:https://brainly.com/question/13278705

#SPJ1

The augmented matrix for the linear system corresponding to the matrix equation Ax = b is |1 2 4 0| |-2 2 9| |0 1 4 1| |2 -1| |-2 -4 -3 -1| |9 -1|. Solving the system yields the solution vector (x, y, z) = (5, -1, 4).

The augmented matrix for the linear system Ax = b is a way of combining the two matrices into one matrix. In this case, it is the combination of the coefficient matrix A and the constant matrix b. This new matrix contains all of the information needed to solve the linear system, as it contains both the coefficients of the variables and the solutions of the equations. We can solve the linear system by performing row operations to the augmented matrix, which is equivalent to solving the equations in the original matrix equation. After performing the row operations, we will be left with the solution vector (x, y, z) = (5, -1, 4). This solution vector represents the values of the variables that satisfy the original equation.

Learn more about matrix here

https://brainly.com/question/28180105

#SPJ4

Answer: The equations that represent circles that have a diameter of 12 units and a center that lies on the y-axis are:

(x – 6)² + y² = 36

x2 + (y + 6)² = 36

Because a diameter is the distance between two points on a circle that is farthest from each other, the radius of the process must be half of the diameter. So the radius of the ring is 6 units.

The center of the circle must be on the y-axis, it means that the x coordinate of the center is 0, so the equation of the circle will be (x - h)² + (y - k)² = r² where h = 0, k = center on the y-axis, and r = 6.

The first equation, x2 + (y – 3)2 = 36, doesn't represent a circle with a center on the y-axis, because the center of the circle is represented by (0, 3) which is not on the y-axis.

The second equation x2 + (y – 5)2 = 6 doesn't represent a circle with a diameter of 12 units, because the value of the equation is 6 which is not equal to the square of the radius.

The third equation, (x – 4)² + y² = 36, doesn't represent a circle with a center on the y-axis, because the center of the circle is represented by (4, 0) which is not on the y-axis.

The forth equation, (x + 6)² + y² = 144, doesn't represent a circle with a diameter of 12 units, because the value of the equation is 144 which is not equal to the square of the radius.

The fifth equation, x2 + (y + 8)² = 36, doesn't represent a circle with a diameter of 12 units, because the value of the equation is 36 which is not equal to the square of the radius.

Step-by-step explanation:

Correct option is C, The height of triangular prism is not 6 units, it is 15 units.

A triangle base, a translated copy, and three faces that join the appropriate sides make up a triangular prism, a three-sided prism. There are no rectangular sides in an oblique right triangle; it is only a right triangle. A right triangle is a uniform triangular prism, which has square sides and equilateral bases.

In essence, it is a polyhedron with three parallel sides, three surface normals in the same plane, and three surface normals (which is not necessarily parallel to the base planes). These three faces make it a parallelogram. Every cross-section that follows the base faces yields the same triangle.

For the given triangular prism,

The perimeter of base = Side + Side + Side = 13+11+8 = 32 units

The area of base = 1/2 × Base × Height = 1/2 × 11 × 6 = 33 units²

The height of prism = 15 units

The lateral surface area = (side + side + side) × height = 15 × (13 + 11 + 8) = 480 units²

To Learn more About  triangular prism refer To:

brainly.com/question/23178481

#SPJ1

One example of a nonempty subset u of R^2 such that u is closed under scalar multiplication, but u is not a subspace of R^2, is the set u = {(x, y) | x, y are rational numbers}.

This set is closed under scalar multiplication as if a and (x,y) are in u, a*(x,y) will also be in u because rational numbers are closed under multiplication.

A subset can be described with different properties, one of them is closure, a subset U is closed under an operation if the operation is applied to any two elements of U, and the result is also an element of U.

In mathematics, a subset is a set that is completely contained within another set. It is defined as a set U that is a part of a set S, such that every element of U is also an element of S. The notation used to indicate that U is a subset of S is U ⊆ S.

However, this set u is not a subspace of R^2 because it doesn't contain the zero vector (0,0) and it doesn't close under addition, as the sum of two rational numbers may not be a rational number.

Therefore, One example of a nonempty subset u of R^2 such that u is closed under scalar multiplication, but u is not a subspace of R^2, is the set u = {(x, y) | x, y are rational numbers}.

To know more about subsets refer to:

brainly.com/question/23454979

#SPJ4

Using Pythagorean theorem, AD = 9 units and AC = 25 units.

ΔABC is a right triangle.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Using Pythagorean theorem:

AB² = AD² + BD²

AD = √(AB² - BD²)

AD = √(15² - 12²)

AD = 9 units

AC = AD + CD

AC = 9 + 16 = 25 units

For ΔABC:

The hypotenuse is AC = 25.

Using  Pythagorean theorem to confirm if ΔABC a right triangle. That is:

AC² = BC² + AB²

25² = 20² + 15²

625 = 625

Since the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Therefore, ΔABC is a right triangle.

Learn more about Pythagorean theorem on:

brainly.com/question/343682

#SPJ1

Answer:

Step-by-step explanation:

Let x=child number

Let y=adult number

so,    x+y=143

and  6x+9.5y=1089

Form x+y=143, we got x=143-y

so, 6x+9.5y=1089

will be 6(143-y) +9.5y=1089

then we got y=66

We are looking for adult ticket were sold, so it is 66 tickets.

a. It is more likely to randomly select square yards of material and finding an average of or more flaws.

b. Normal distribution assumes that the data is continuous and symmetric, but in this case, the number of flaws per square yard is a count data, and it is not symmetric. Therefore, a normal distribution cannot be used to calculate the probability of the first event in part (a).

c. To calculate the probability of the second event in part (a), we would need the mean and standard deviation of the number of flaws per square yard, as well as the specific threshold of "or more flaws" to calculate the area under the curve that is greater than or equal to that threshold. Without that information, I cannot provide an accurate probability calculation.

I determined that it was not possible to calculate the probability of the second event in part (a) without the specific mean and standard deviation of the number of flaws per square yard and the specific threshold of "or more flaws." This is because in order to calculate the probability of an event, you need to know the underlying distribution of the data and the specific cut-off point for the event in question.

In the other part of the question, I stated that it is more likely to randomly select square yards of material and finding an average of or more flaws than it is to randomly select a square yard of material and finding 2 or more flaws.

I came to this conclusion without any calculations because in general, when you are looking for a specific number of flaws (in this case, 2 or more flaws) in a single square yard, it is typically less likely than finding an average of that number of flaws when you look at multiple square yards. This is because the number of flaws per square yard can vary, so even if the average is 2 or more flaws per square yard, it is still possible to find square yards with fewer than 2 flaws.

Learn more about Carpet Flaw Probability Calculation here:

https://brainly.com/question/1698123

#SPJ4

Answer:

marginal revenue = $21

Monopolistic competition is a market structure where many firms sell differentiated products. The monopolistic  competitive firm maximizes profits where marginal revenue is above or equals to marginal cost. Profit is maximum at an output of 4 units. When the output is 4 units, marginal revenue is $21 and marginal cost is $7.  

The biggest number of children in the group would be 14 boys and 14 girls.

Basic division is an arithmetic operation that involves two numbers. It is used to find out how many times one number can be divided into another. It is sometimes referred to as division or division by a divisor.

Division is the inverse operation of multiplication. This means that division is the opposite of multiplication. When a number is multiplied by another number, the result is the product of the two numbers. When a number is divided by another number, the result is the quotient of the two numbers.

The process of division involves finding the number of times that one number can be divided into another. Division is performed by dividing the dividend (the number being divided) by the divisor (the number that divides the dividend). The result of the division is the quotient.

Division can be done using a variety of methods. Long division is a traditional method of division, often taught to students in elementary school. It involves breaking down each number into individual digits and then dividing them one at a time.

Another type of division is short division, which is a simpler and faster method of division that can be done in one step. Short division can be used when the divisor is a single-digit number.

To learn more about basic division, visit

brainly.com/question/941863

#SPJ1

Note that the vertical rise of the included plane is 519.35 ft. This is arrived at by the use of the knowledge of Trigonometric Identities.

Trigonometric Identities are equalities that utilize trigonometry functions and hold true for all variables in the equation. There are several trigonometric identities relating the side length and angle of a triangle.

Since the Johnstown Inclined Plane in Pennsylvania is known to be one of the world's longest and steepest hoists. The railway trains traverse 896.5 feet at an inclination of around 35.4 degrees, reaching a height of 1693.5 feet above sea level.

Because the slope's length is 896.54 feet and its inclination is 35.4 degrees, let h be the vertical rise of the incline. Based on trigonometric identities,

Sin 35.4 = h/896.54

h = 896.54 (Sin35.4)

h = 519.35 feet.

Thus, it is right to state that the vertical rise of the included plane is 519.35 ft.

Learn more about Trigonometric Identities:
https://brainly.com/question/24377281
#SPJ1

Full Question:

The Johnstown Inclined Plane in Pennsylvania is one of the longest and steepest hoists in the world. The railway cars travel a distance of 896.5 feet at an angle of approximately 35.4 degrees, rising to a height of 1693.5 feet above sea level. Find the vertical rise of the inclined plane.

(Round your answers to two decimal places.)

The reduced integral form of the equation x = 4 tan θ is 4ln|sec x| + C

The term integral is referred as the process to make use of different simplification methods depending upon the type of the integrand.

Here we have the equation x = 4 tan θ

Here in this problem, we have to make use of the substitution method to simplify the integrals.

Here we have to apply these value on the integral form then we get,

=> ∫x dx

=> ∫4 tan θ dθ

As we all know that when we integrate we get the resulting value as,

Here the number 4 is a constant, so it can be written as,

=> 4  ∫tan θ dθ

Then the resulting value is written as,

=> 4ln|sec x| + C

To know more about integral here.

https://brainly.com/question/18125359

#SPJ4

Answer: To determine which expression is equivalent to 3√/8x²y³ z^4, we need to simplify the given expression and compare it with the options.

First, we can simplify the cube root:

3√/8x²y³ z^4 = (3/8)x^(2/3)y^(1/3)z^(4/3)

Now, we can compare this expression with the options:

A. 2x^(3/2)YZ^(3/4) is not equivalent to the simplified expression because the exponent of x is different (3/2 vs 2/3)

B. 2x^(2/3)YZ^(3/4) is equivalent to the simplified expression. The exponents of x, y and z are the same and the coefficient is also the same

C. 2z/x is not equivalent to the simplified expression because it has different variables and exponents

D. 2x/z is not equivalent to the simplified expression because it has different variables and exponents

Therefore, the expression that is equivalent to 3√/8x²y³ z^4 is B. 2x^(2/3)YZ^(3/4)

Step-by-step explanation:

An equation of the line of best fit is y = 0.97x + 1.67.

The plant is expected to be 14.28 units tall after 13 months.

In order to determine a linear equation for the line of best fit (trend line) that models the data points contained in the table, we would have to use a scatter plot.

In this scenario, the number of months would be plotted on the x-coordinate of the scatter plot while the height would be plotted on the y-coordinate of the scatter plot.

On the Excel worksheet, you should right click on any data point on the scatter plot, select format trend line, and then tick the box to display an equation for the line of best fit (trend line) on the scatter plot.

From the scatter plot (see attachment) which models the relationship between the number of months and the height, a linear equation for the line of best fit is given by:

y = 0.97x + 1.67

Next, we would determine the predicted height of the plant after 13 months:

y = 0.97(13) + 1.67

y = 14.28 units.

Read more on scatter plot here: brainly.com/question/28605735

#SPJ1

Answer:

(-5,-37)

Step-by-step explanation:

Complete the square

(x+5)^2-37

a) The sampling method used is non-probability sampling.

b) The sample is going to be representative.

What is sampling?

Sampling is a technique for choosing certain individuals or a small portion of the population in order to draw conclusions about the population as a whole and estimate its characteristics. Researchers frequently utilise various sampling techniques in market research so they do not have to study the full community in order to gather useful information.

In non-probability sampling, participants are chosen at random by the researcher. This type of sampling is not a set or predetermined selection procedure. This makes it challenging for every component of the population to have an equal chance of being included in a sample.

The sampling method used in non-probability sampling as the people selected are selected on a convenience basis.

A representative sample is a subset of a population that seeks to accurately reflect the characteristics of the larger group.

Since, the students are chosen randomly but from the same room of the same school they might represent the characteristics of all other students.

Therefore, the sample is representative.

To learn more about sampling from the given link

https://brainly.com/question/24466382

#SPJ1

A changelog is a record of changes made to a dataset during the data cleaning process.

It can serve as a helpful reference for data analysts and other stakeholders to understand the changes that were made and the reasons why they were made. It is important for data analysts to keep a detailed changelog in order to learn from their mistakes and to provide a detailed audit trail for any future questions about the data.

The formula for calculating the value of a changelog is as follows:

changelog= Number of Changes Made + Complexity of Changes + Accuracy of Changes

The number of changes made to the data set is an important factor in calculating the changelog value. The more changes that are made, the more valuable the changelog will be. The complexity of the changes is also important. If the changes are complex and require a lot of effort to make, the changelog will be more valuable. Finally, the accuracy of the changes is also a factor. If the changes are accurate and correct, then the changelog will be more valuable.

Overall, a changelog is a valuable tool for data analysts to keep a detailed record of the changes they make to a dataset. This record can be used to ensure accuracy and to provide a detailed audit trail for any questions that may arise in the future.

Learn more about  factor here:

https://brainly.com/question/29128446

#SPJ4

The amount of metal does artist need to make the sculpture is 990,845.54 cm²

Step 1: Calculate the total surface area of the rectangular prism.

Surface area = 2(wl + hl + hw)

Surface Area = 2((21.8 x 24.4) + (21.8 x 39.6) + (24.4 x 39.6)) = 4722.88

Step 2: Calculate the total volume of the rectangular prism.

V = l * w * h *

Volume = 21.8 x 24.4 x 39.6 = 21064.03 cm³

Step 3: Multiply the total surface area by the total volume to get the total amount of metal needed.

Total Metal Needed = 4722.88 cm² x  21064.03 cm³ = 990,845.54cm²

To learn more about surface area refer to:

https://brainly.com/question/1297098

#SPJ1

The requried expression that represents a 95% confidence interval for the slope of the least squares regression line is:

b₁ ± t₁-α/2, n-2 * se(b₁)

To calculate a 95% confidence interval for the slope of the least squares regression line, we can use the following formula:

95% C.I. for β₁: b₁ ± t₁-α/2, n-2 * se(b₁)

where:

b₁ = Slope coefficient is shown in the regression table.

t₁-α/2, n-2 = The t critical value for confidence level 1-α with n-2 degrees of freedom where n is the total number of observations in our dataset.

se(b1) = The standard error of b₁ shown in the regression table.

Given that all conditions for inference are met, we can use the output of the linear regression analysis to find the values of b1 and se(b1) and then use a t-distribution table to find the t critical value for a 95% confidence level with 20 degrees of freedom (22-2).

The given options do not provide the values of b1 and se(b1), so we cannot calculate the confidence interval using them. However, we can use the formula and the information provided in the search results to understand how to calculate the confidence interval for the slope of the least squares regression line.

Therefore, the expression that represents a 95% confidence interval for the slope of the least squares regression line is:

b₁ ± t₁-α/2, n-2 * se(b₁)

where b₁ and se(b₁) are the slope coefficient and the standard error of the slope coefficient, respectively, obtained from the regression table, and t₁-α/2, n-2 is the t critical value for a 95% confidence level with n-2 degrees of freedom, obtained from a t-distribution table.

Learn more about regression lines here :
https://brainly.com/question/31648883

#SPJ4

Comparing column a with column b we get that column b is greater than column a. Hence, option B is the correct answer to this question.

Column a: We can find the surface area of a cone by using the formula -

Surface area = π * r * (r + l)

where r is the radius of the base and l is the slant height of the cone.

Substituting r = 3 cm and l = 8 cm, we get,  

Surface area = π * 3 * (3 + 8) = π * 3 * 11 = 33π

Hence, Surface area = 33π  [tex]cm^2[/tex]

Column b: We can find the surface area of a cone by using the formula -

Surface area = π * r * (r + l)

where r is the radius of the base and l is the slant height of the cone.

Substituting r = 4 cm and l = 6 cm, we get,

Surface area = π * 4 * (4 + 6) = π * 4 * 10 = 40π

Hence, Surface area = 40π  [tex]cm^2[/tex]

Comparing we get that,

The surface area of the cone with a base radius of 3 cm and slant height of 8 cm < the surface area of the cone with a base radius of 4 cm and a slant height of 6 cm.

Hence, column a < column b.

Thus, the correct option is B. The quantity in column B is greater.

Read more about cone:

https://brainly.com/question/30253224

#SPJ4

The complete question is -

Compare the quantity in column a with the quantity in column b. choose the best answer.

column a: The surface area of the cone with a base radius of 3 cm and a slant height of 8 cm.

column b: The surface area of the cone with a base radius of 4 cm and a slant height of 6 cm.

A. The quantity in column A is greater.

B. The quantity in column B is greater.

C. The two quantities are equal.

D. The relationship cannot be determined from the information given.

Answer:

35/12

Step-by-step explanation:

In a right angled triangle tangent is the ratio of perpendicular and base . In the given triangle

So ,

[tex]\longrightarrow \tan\alpha =\dfrac{p}{b} \\[/tex]

[tex]\longrightarrow \underline{\underline{ \tan\alpha =\dfrac{35}{12} }} \\[/tex]

And we are done!

Answer:

We multiply 3/4 with the reciprocal of 3/8 which is 8/3. This leaves us with 3/4x8/3 which equals 24/12. This can be simplified into 2.

In order yo calculate the perimeter of a triangle, one should add the length of its sides. For example, if the triangle has sides a, b, and c, then the perimeter of that triangle will be P = a + b + c

Any two-dimensional figure's perimeter is determined by the space surrounding it. By adding the lengths of the sides, we can determine the perimeter of any closed shape.

The perimeter of any polygon is equal to the sum of its side lengths. Considering a triangle, sum of the three sides equals the perimeter.

Units are always a part of the final response. The final answer should be given in centimeters if the triangle's sides are in that unit.

An overview was given as your information is incomplete and couldn't be found.

Learn more about triangles on:

https://brainly.com/question/24382052

#SPJ1

We need to use the substitution method of integration and logarithm properties to provide the statements.

We know that

cotx = cosx/sinx

∫cotx dx = ∫[cosx/sinx]dx

Let sinx be z

differentiating with respect to x we get

cosx dx = dz

Using this we get

∫[1/z]dz

We know that ∫1/x dx = logx + C

Hence we get

logz + C

Using the value of z here we get

log|sinx| + C

Here we know that we cannot have log of a negative value hence the mod sign has to be used.

Now,

∫cotx dx = ∫cosx.cosecx dx

let cosecx be z

diff with respect to x we get

-cot x.cosec x dx = dz

or, -cosx.cosec²x dx = dz

or, dx = dz/[-z²cosx]

Hence we get

∫[cosx.z]/[-z²cosx]dz

or, -∫1/z dz

or, -logz + C

or, -log|cosecx| + C

Hence Proved

To learn more about Integration visit

https://brainly.com/question/14405228

#SPJ4

Complete Question

Using Properties of Logarithms and Trigonometric Identities In Exercises 88 - 92 show that the two formulas are equivalent.

∫cotx dx = ln|sinx| + C

∫cotx dx = -ln|cosecx| + C