Exercise 6.2.8: Solve ′′′ + = 3( − 1) for initial conditions (0) = 1 and ′(0) = 0, ′′(0) = 0.

a. From the graph a = 1 and b = 1

b. f(x) = -x/4 + 5

c. The area under the graph is 9 units²

a. First to find the area under the graph, we find the values of a and b. From the graph, a = 1 and b = 3

b. To find f(x), we note that f(x) is a straight line. So, using the equation of a

in slope form, we have that

(y - y')/(x - x') = (y" - y')/(x" - x') where

So, substituting the values of the variables into the equation, we have that

(y - y')/(x - x') = (y" - y')/(x" - x')

(y - 5)/(x - 0) = (4 - 5)/(4 - 0)

(y - 5)/x = -1/4

y - 5 = -x/4

y = -x/4 + 5

So, f(x) = -x/4 + 5

c. To find the area under the curve, we notice that ∫f(x)dx = area

So, ∫f(x)dx = ∫₁³(-x/4 + 5)dx

= ∫₁³(-xdx/4) +  ∫₁³5dx

= [-x²/(2 × 4)]₁³ + [5x]₁³

= [-x²/8]₁³ + [5x]₁³

= [-3²/8 - (-1)³/8] + 5[3 - 1]

= [-9/8 + 1/8] + 5[2]

= -8/8 + 10

= -1 + 10

= 9 units²

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Plug in 0 for 'x' to find the y-intercept:

[tex]7x - 3y = 21[/tex]

[tex]7(0) - 3y = 21[/tex]

[tex]0 - 3y = 21[/tex]

[tex]-3y = 21[/tex]

Divide -3 to both sides:

[tex]y = -7[/tex]

So the y-intercept is -7.

[tex]7x - 3y = 21[/tex]

Plug in 0 for 'y' to find the x-intercept.

[tex]7x - 3(0) = 21[/tex]

[tex]7x - 0 = 21[/tex]

[tex]7x = 21[/tex]

Divide 7 to both sides:

[tex]x = 3[/tex]

So the x-intercept is 3.

The volume of the rectangular prism is 144 in³ with the area of the base is 24 square inches and the height is 6 inches.

Volume is a measure of the amount of physical space occupied by an object or a fluid.

This is because the volume of a rectangular prism can be calculated by the formula V = A x h, where V is the volume, A is the area of the base, and h is the height. In this problem, the area of the base is 24 square inches and the height is 6 inches.

Therefore, the volume of the rectangular prism can be calculated as follows:

V = 24 in² x 6 in

= 144 in³

It is important to note that the formula for calculating the volume of a rectangular prism can be used for any rectangular prism, regardless of the size of the area of the base and the height. The formula remains the same and the only thing that changes is the numbers that are used to calculate the volume.

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The measure of the angle ∠6 is 102 degrees for the given parallel lines.

Two angles are said to be supplementary if their sums equal 180 degrees. In other words, if angles A and B are complementary, then the sum of the two angles is 180 degrees. Four angles are created when two lines intersect; if two of these angles are supplementary, the other two are likewise supplementary.

Since 70 + 110 = 180 degrees, 70 degrees and 110 degrees serve as an illustration of a pair of supplementary angles. Another illustration is a pair of vertical angles, which are always supplementary and are generated by the intersection of two lines.

The measure of angle of a straight line is given as 180 degrees.

Thus,

∠6 + 78 = 180

∠6 = 180 - 78

∠6 = 102

Hence, the measure of the angle ∠6 is 102 degrees for the given parallel lines.

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Answer:

$705.71

Step-by-step explanation:

Let's assume the regular price of the sofa is "x".

According to the problem statement, the sale price is 65% off the regular price, which means the sale price is equal to 35% of the regular price.

We can write this relationship as:

0.35x = $247

To find the value of "x", we need to solve for "x" in the above equation.

Dividing both sides by 0.35, we get:

x = $247 ÷ 0.35

x = $705.71

Therefore, the regular price of the sofa is $705.71.

Answer:

The graphs of these lines intersect at

(0, 3).

Answer:

Hilltop

Step-by-step explanation:

Convert all to meters,

Mammoth- 8005 meters

Brookside- 8700 meters

Wild Rose- 8080 meters

Hilltop- 8800 meters.

Hilltop has the most meters, it is the longest.

Answer:

3:5 or 3 to 5 or [tex]\frac{3}{5}[/tex]

Step-by-step explanation:

[tex]\frac{6}{10}[/tex] ÷[tex]\frac{2}{2}[/tex] = [tex]\frac{3}{5}[/tex]

Helping in the name of Jesus.

The final product of the given algebraic expression is: b⁴ - 64

Algebraic expressions are defined as simply the idea of expression of numbers using letters without specifying their actual values. The basics of algebra taught us how to express an unknown value using letters such as a, b, c, etc. These letters are referred to as variables. An algebraic expression can be a combination of both the variables and the constants. Any value that is placed before and multiplied by a variable is a coefficient.

The expression is given as:

(b² + 8)(b² - 8)

Expand the expression using (a + b)(a - b) = a² - b²

Thus:

(b² + 8)(b² - 8) = b⁴ - 64

Thus, that is the final product of the given algebraic expression.

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Answer:

x = 6.111... and y = 1.

Step-by-step explanation:

To use Cramer's Rule, we first write the system of equations in matrix form, where the coefficients of x and y are placed in a matrix, and the constant terms are placed in a separate matrix. We then calculate the determinant of the coefficient matrix, which is the number that is obtained by multiplying the diagonal elements and subtracting the product of the off-diagonal elements.

Next, we replace the first column of the coefficient matrix with the constant terms and calculate the determinant of the resulting matrix. We do the same thing for the second column of the coefficient matrix. We then use Cramer's Rule to solve for x and y, which involves dividing the determinant of the matrix obtained by replacing the column of x coefficients with the constant terms by the determinant of the coefficient matrix to get the value of x. We do the same thing to find the value of y by dividing the determinant of the matrix obtained by replacing the column of y coefficients with the constant terms by the determinant of the coefficient matrix.

In the case of the given system of equations, the determinant of the coefficient matrix is 18. We then calculate the determinants of the matrices formed by replacing the first column of the coefficient matrix with the constant terms and the second column of the coefficient matrix with the constant terms. Using these determinants, we apply Cramer's Rule to find the values of x and y, which are x = 6.111... and y = 1.

Answer:

4.5

Step-by-step explanation:

uh

The line an illustrates a linear function. Line a's slope is thus 1. Y = x + 2 is the equation for the line 'a'.

The steepness or slant of a line may be determined by looking at its slope. The slope is defined as the ratio of any two points on the line's vertical change (rise) to its horizontal change (run). The letter "m" is typically used to indicate slope.

The slope of a line connecting the coordinates (x1, y1) and (x2, y2) may be determined mathematically using the following formula:

[tex]m=\frac{y_{2}-y_{1} }{x_{2} -x_{1} }[/tex]

Now, Evaluating slope of line 'a';

[tex]m=\frac{4-2 }{2-0 } = \frac{2}{2} =1[/tex]

Hence, the slope of line 'a' is 1.

Final Plotted Graph is mentioned in image 3 attached.

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Complete Question : Draw a vertical y-axis on the left-hand side of your graph and label it. Then draw a horizontal x-axis at the bottom of your graph. Then label the coordinates of each corner of the sign on your graph(Refer to image attached). Draw line a on your graph. What is the slope of line a?

Union will need to choose a random sample of at least 65 wages to be 90% confident that its estimate of the mean hourly wage of foodservice workers is within $0.45 of the true population mean.

To calculate the minimum sample size needed for the union to be 90% confident within $0.45 of the true population mean, we can use the formula which states n = (Z^2 * σ^2) / E^2

Z = 90% = 1.645

σ = $2.20

E = $0.45

Substituting the values, we get:

n = (1.645^2 * 2.20^2) / 0.45^2

n = 13.097161 / 0.2025

n = 64.6773382716

n = 64.678

n = 65

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Answer:

x = [tex]\frac{65}{17}[/tex]

Step-by-step explanation:

[tex]\frac{x}{13}[/tex] = [tex]\frac{5}{17}[/tex] ( cross- multiply )

17x = 13 × 5 = 65 ( divide both sides by 17 )

x = [tex]\frac{65}{17}[/tex] ≈ 3.8 ( to the nearest tenth )

Answer:

it's for sure d please mark me as a brainliest

Answer:

170.52

Step-by-step explanation:

Let x be the original price

Then,

129.60 = 76% of x

129.60 = 76/100 * x

12960/76 = x

x = 170.52

based on the information provided, the probability of selecting a yellow disk is 1/5 or 0.2.

How to solve the problem?

The probability of selecting a yellow disk cannot be 2, as probabilities must always be between 0 and 1.

To calculate the probability of selecting a yellow disk, we can use the formula:

P(yellow) = number of yellow disks / total number of disks

From the information given, we know that there are 5 yellow disks and 25 total disks in the bag. Plugging these values into the formula gives:

P(yellow) = 5/25 = 1/5

Therefore, the probability of selecting a yellow disk is 1/5 or 0.2, which is a decimal value between 0 and 1.

It's possible that there was a typo in the original question or answer, or that some other context is missing which would make sense of the answer being 2. However, based on the information provided, the probability of selecting a yellow disk is 1/5 or 0.2.

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Answer:

x = 3

y = 23

z = 4

Step-by-step explanation:

To find the values of x, y, and z in the rhombus, we need to use the properties of a rhombus. One of the properties of a rhombus is that its opposite angles are equal. Another property is that its diagonals are perpendicular bisectors of each other.

Let's label the vertices of the rhombus as A, B, C, and D, and the midpoint of diagonal AC as M. We can use the given information to write some equations:

First, we know that diagonal AC is perpendicular to diagonal BD, so we have:

-2x + 6 = 0

Solving for x, we get:

x = 3

Next, we know that diagonal AC bisects angle ACD, so we have:

y = 46/2 = 23

Now, we can use the fact that AM is the perpendicular bisector of CD to find z. Since M is the midpoint of AC, we have:

AM = MC

Using the distance formula, we can find the lengths of AM and MC in terms of z:

AM = sqrt((2z-8)^2 + (5y-4)^2)

MC = sqrt((8-2z)^2 + (5y-4)^2)

Setting these two expressions equal to each other, we get:

sqrt((2z-8)^2 + (5y-4)^2) = sqrt((8-2z)^2 + (5y-4)^2)

Simplifying this equation, we get:

4z - 16 = 16 - 4z

8z = 32

z = 4

Therefore, the values of x, y, and z are:

x = 3

y = 23

z = 4

So, the solution is x=3, y=23, and z=4.

Answer: check explanation

Step-by-step explanation:

Since ABCD is a parallelogram, its opposite sides are parallel and congruent. Therefore, we have:

AB || CD and AB ≅ CD

AD || BC and AD ≅ BC

Since AC is a diagonal, it divides the parallelogram into two congruent triangles, namely, ΔABC and ΔACD. Therefore, we have:

∠A ≅ ∠D (corresponding angles of congruent triangles)

∠C ≅ ∠B (corresponding angles of congruent triangles)

AB ≅ CD (opposite sides of parallelogram)

AC ≅ AC (reflexive property of congruence)

Now, consider the triangles ΔAEC and ΔCEB. We have:

∠AEC ≅ ∠CEB (vertical angles)

∠ACE ≅ ∠BCE (corresponding angles of congruent triangles ΔABC and ΔACD)

AC ≅ AC (reflexive property of congruence)

Therefore, by the angle-angle-side (AAS) postulate, we can conclude that ΔAEC ≅ ΔCEB. Hence, we have:

CE ≅ AC (corresponding parts of congruent triangles)

BE ≅ DE (corresponding parts of congruent triangles)

Thus, we have proven that line segment AC is congruent to line segment CE, and line segment BE is congruent to line segment DE.

The average daily balance is 163.904.

Given that Edie Bertelli received her statement from a department store.

A) The average daily balance =

sum of daily balance / no. of days = 4917.12/30 = 163.904.

B) Finance charges =  

average daily balance x period rate = 163.904 x 1.20% = 1.966%

C) New balance =

(Opening balance + finance charge + new purchase) - payment =

(175 + 1.966 + 55.94) - 90 = 142.9068

Hence, the average daily balance, the finance charges and the new balance is 163.904, 1.966 and 142.908 respectively.

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Sarah's daily outgoings on wages are £892.50.

What is wages?

Wages refer to the payment made to an employee in exchange for their labor or services provided to an employer. It is usually paid on an hourly, daily, weekly or monthly basis, and the amount of wages earned is determined by the hours worked, rate of pay, and any applicable deductions or taxes. Wages can be paid for various types of work, such as manual labor, skilled trades, clerical work, or professional services. It is an essential component of an employee's compensation package and is regulated by labor laws to ensure that employees are fairly compensated for their work.

Each employee works for 8.5 hours a day (17:00 - 08:30 = 8.5). So, the total number of hours worked by 12 employees in a day is 12 employees × 8.5 hours/employee = 102 total hours

Sarah pays each employee £8.75 per hour, so her total daily outgoings on wages can be calculated as,

102 total hours × £8.75 per hour/employee = £892.50

Therefore, Sarah's daily outgoings on wages are £892.50.

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Answer:

X = -8 X= -4

Step-by-step explanation:

x² + 12x = -32

(x + 12/2)² = -32 + (12/2)²

(x + 6)²= 4

square root both sides

x + 6 = 2

x + 6 = -2

x=-8 x=-4

Answer:

1. 16

2. 9

3. 21.08

Step-by-step explanation:

1. (explanation) (x+b)^2 = x^2+2bx + b^2 ( square formula)

16, would make a square trinomial

Answer:

1.16

2.9

3.x=9+root of 146(12.08) or x=9-root of 146

Rachel $8500 invested at 4% and $8800 invested at 2% interest rate.

Investment is the act of allocating money or resources with the expectation of generating income or profit in the future.

Let x be the amount invested in the second account (with a 4% interest rate).

Then, since Rachel invested $300 more in the first account, the amount invested in the first account (with a 2% interest rate) is (x + 300)

We know that the total interest income Rachel earned was $516. We can set up an equation using the interest formula: (2% means 0.02 and 4% means 0.04)

⇒ 0.02(x + 300) + 0.04x = 516
⇒ 0.02x + 6 + 0.04x = 516
⇒ 0.06x + 6 = 516
⇒ 0.06x = 510
⇒ x = 8500

So, Rachel invested $8500 in the second account (with a 4% interest rate)

and (x + 300) = (8500 + 300) = $8800 in the first account (with a 2% interest rate).

Therefore, Rachel $8500 invested at 4% and $8800 invested at 2%.

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If Mike scores two standard deviations above the mean then the Mike's percentile-score is 97.72 ≈ 98 percentile.

The percentile score corresponding to a raw test score, is the probability of the test score of a randomly chosen student to be less than or equal to the given test score.  To calculate this probability we find the value of the cumulative distribution function of the test score random variable at the given test score.

To calculate the percentile score of the given raw test score we have to find the value of the cumulative distribution function for the test score random variable at the given raw test score. if this random variable is X, then the linearly scaled random variable [tex]Z=\frac{X-\mu}{\sigma}[/tex], has standard normal-distribution. here [tex]\mu\textrm{ and }\sigma[/tex] are the mean and the standard deviation of the test score r.v respc. Also [tex]\textrm{P}(X\leq x_0) = \textrm{P}(Z\leq z_0),\textrm{ where }z_0=\frac{x_0-\mu}{\sigma}[/tex].  So instead of finding [tex]\textrm{P}(X\leq x_0)\textrm{, we can find } \textrm{P}(Z\leq z_0)[/tex] instead. [tex]z_0[/tex] is called the z-score corresponding to the [tex]x_0[/tex]. We can find [tex]\textrm{P}(Z < z_0)[/tex], using the table for the standard normal distribution.

For our question, we want to find [tex]\textrm{P}(X\leq \mu + 2\sigma)[/tex]. which is equal to [tex]\textrm{P}(Z\leq \frac{\mu+2\sigma - \mu}{\sigma}) = \textrm{P}(Z\leq 2).[/tex] So the z-score = 2. From the table for the standard normal distribution we get [tex]\textrm{P}(Z\leq 2) = .9772[/tex]. So our percentile score is .9772 or 97.72% [tex]\sim[/tex] 98%.

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The answer of the given question based on the trigonometric identities is , A. sin(-Θ) = -B , b. tan(-Θ) = -A , C. cos(Θ + 2π) = B/√(A² + B²) , D. tan(Θ + π) = -A.

Trigonometric identities are equations in mathematics that use trigonometric functions and hold true regardless of the value of the variables. These identities can be applied to simplify or alter trigonometric expressions, as well as to trigonometric function-based equations.

To determine the required values, we can utilise sine and tangent definitions along with trigonometric identities:

a. sin(-Θ) = -sin(Θ) = -B (using the property that sine is an odd function)

b. tan(-Θ) = -tan(Θ) = -A (using the property that tangent is an odd function)

c. cos(Θ + 2π) = cos(Θ) = B/√(A² + B²) (using the Pythagorean identity, since cos²(Θ) + sin²(Θ) = 1)

d. tan(Θ + π) = (tan(Θ) + tan(π))/(1 - tan(Θ)tan(π) = (-A + 0)/(1 + A0) = -A

Therefore, the values are:

a. sin(-Θ) = -B

b. tan(-Θ) = -A

c. cos(Θ + 2π) = B/√(A² + B²)

d. tan(Θ + π) = -A

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The complete question is:

If tan (θ) = A, sin(θ)= B,

Then find the following:

a. sin(-θ)=

b. tan(-θ)=

c. cos(θ+ 2π) =

d. tan(θ+ π)=

The answer of the given question based on the trigonometric identities is , A. sin(-Θ) = -B , b. tan(-Θ) = -A , C. cos(Θ + 2π) = B/√(A² + B²) , D. tan(Θ + π) = -A.

Trigonometric identities are equations in mathematics that use trigonometric functions and hold true regardless of the value of the variables. These identities can be applied to simplify or alter trigonometric expressions, as well as to trigonometric function-based equations.

To determine the required values, we can utilise sine and tangent definitions along with trigonometric identities:

a. sin(-Θ) = -sin(Θ) = -B (using the property that sine is an odd function)

b. tan(-Θ) = -tan(Θ) = -A (using the property that tangent is an odd function)

c. cos(Θ + 2π) = cos(Θ) = B/√(A² + B²) (using the Pythagorean identity, since cos²(Θ) + sin²(Θ) = 1)

d. tan(Θ + π) = (tan(Θ) + tan(π))/(1 - tan(Θ)tan(π) = (-A + 0)/(1 + A0) = -A

Therefore, the values are:

a. sin(-Θ) = -B

b. tan(-Θ) = -A

c. cos(Θ + 2π) = B/√(A² + B²)

d. tan(Θ + π) = -A

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The complete question is:

If tan (θ) = A, sin(θ)= B,

Then find the following:

a. sin(-θ)=

b. tan(-θ)=

c. cos(θ+ 2π) =

d. tan(θ+ π)=

Answer: The value of x will be equal to 5.

Step-by-step explanation:

1. Since all the angles of the given triangle is equal it proves to be an Equilateral Triangle.

2. Now as all the angles are equal the length of side will be equal too.

3. Equating both equations: 3x-3 = 2x+2

3x-2x = 3+2 => x = 5.

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According to the question there are 35 distinct website graphics that can be created by using at least four of seven different bitmap images.

A website is a collection of related web pages, images, videos, and other digital assets that are hosted on a web server and can be accessed by users over the internet. A website is typically identified by a unique domain name and can be accessed by typing the URL into a web browser.

Using the formula for calculating the number of combinations of size k from a set of size n (n choose k), we can calculate the number of distinct website graphics that can be created by using at least four of seven different bitmap images as follows:

n = 7 (7 different bitmap images)
k = 4 (at least 4 of the 7 bitmap images)
Number of distinct website graphics = (7 choose 4) = 35
Therefore, there are 35 distinct website graphics that can be created by using at least four of seven different bitmap images.

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