Kate buys a greeting card for 3.79. She then buys 4 postcards that all cost the same amount. The total cost is 5.11. How much is each postcard? Show your work.

The function of the power series f(x) = 6ln(1+x) = ∑=1 , ∞ 6(-1)⁽ⁿ⁻¹⁾xⁿ/n.

And a_n = 6(-1)⁽ⁿ⁻¹⁾xⁿ/n.

The common ratio is the distance between each number in a geometric series. The proportion of a number or two consecutive numbers. The common ratio, which is the same for all numbers or common, is the number divided by the number that comes before it in the sequence.

To find the power series for f(x), we first need to find the power series for g(x) = 6/(1+x):

g(x) = 6/(1+x) = 6(1 - x + x² - x³ +...) (geometric series with common ratio -x)

Next, we integrate term by term:

∫ g(x) dx = ∫ 6(1-x+x²-x³+...) dx

= 6(x - x²/2 + x³/3 - x⁴/4 + ...) + C , where C is a constant.

Since we're only interested in finding the coefficients of the power series, we can ignore the constant term.

Thus, the power series for f(x) is:

f(x) = 6ln(1+x) = 6(x - x²/2 + x³/3 - x⁴/4 + ...) = ∑=1 , ∞ 6(-1)⁽ⁿ⁻¹⁾xⁿ/n

where a_n = 6(-1)⁽ⁿ⁻¹⁾xⁿ/n.

Therefore, f(x) = 6ln(1+x) = ∑=1 , ∞ 6(-1)⁽ⁿ⁻¹⁾xⁿ/n.

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The sample size needed to ensure that the probability that the sample mean differs from the population mean by more or less then 2.0 hours is less than at a 90% confidence level is 127.

The sample size needed to ensure that the probability can be calculated using the formula:
n = (zα/2)2 2 / (E2)

Where n is the sample size, is the population standard deviation (12.4 hours in this case), E is the margin of error (2.0 hours in this case) and zα/2 is the critical value from a z-table corresponding to the 90% confidence level (1.645 in this case).

Thus, n = (1.645)2 (12.4)2 / (2.0)2 = 126.8

Therefore, the sample size needed to ensure that the probability that the sample mean differs from the population mean by more or less then 2.0 hours is less than at a 90% confidence level is 127.

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If F(-9,-4), E(-7, a) G(-2,5), and H(-6,-1) then the value of a is equal to -1, then EF is parallel to GH.

What are the parallel lines?

Parallel lines are lines in a plane that are always the same distance apart. Parallel lines never intersect.

To determine if EF is parallel to GH, we need to compare the slopes of the two lines.

The slope of line EF can be found using the coordinates of points E and F:

slope of EF = (y2 - y1)/(x2 - x1) = (a - (-4))/(-7 - (-9)) = (a + 4)/2

The slope of line GH can be found using the coordinates of points G and H:

slope of GH = (y2 - y1)/(x2 - x1) = (5 - (-1))/(-2 - (-6)) = 6/4 = 3/2

For EF to be parallel to GH, their slopes must be equal. Therefore, we can set the slope of EF equal to the slope of GH and solve for a:

(a + 4)/2 = 3/2

Multiplying both sides by 2, we get:

a + 4 = 3

Subtracting 4 from both sides, we get:

a = -1

Therefore, if a is equal to -1, then EF is parallel to GH.

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Answer:

approximately 10.3 days to reach 500 bacteria

Step-by-step explanation:

Let's use the formula for exponential growth to write an equation that represents the situation: N = N0 * e^(rt)

where N is the number of bacteria, N0 is the initial number of bacteria, e is Euler's number (approximately 2.718), r is the growth rate, and t is the time in days.

We know that after 4 days, the number of bacteria is 71, so we can plug these values into the equation to solve for the growth rate: 71 = N0 * e^(4r)

Similarly, after 7 days (4 + 3), the number of bacteria is 185: 185 = N0 * e^(7r)

Now we have two equations with two unknowns (N0 and r). We can divide the second equation by the first equation to eliminate N0: 185/71 = e^(3r)

Taking the natural logarithm of both sides, we get: ln(185/71) = 3r

Solving for r, we get: r = ln(185/71) / 3 ≈ 0.558

Now we can use the first equation and the growth rate we just found to solve for N0:

71 = N0 * e^(4 * 0.558)

N0 ≈ 11.7

So the initial number of bacteria was approximately 11.7.

To find out how long it will take to reach 500 bacteria, we can plug in the values we know into the equation and solve for t: 500 = 11.7 * e^(0.558t)

Dividing both sides by 11.7, we get: e^(0.558t) ≈ 42.74

Taking the natural logarithm of both sides, we get: 0.558t ≈ ln(42.74)

Solving for t, we get: t ≈ ln(42.74) / 0.558 ≈ 10.3 days

Therefore, it will take approximately 10.3 days for the sandwich to be fully covered with bacteria.

The equation that links x and y is [tex]y=(\frac{1}{3} )x+3[/tex] if x is 3,6,9,12,15 & y is 4,5,6,7,8.

The definition of the an equation in algebra is a mathematical assertion that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign.

A mathematical equation is a statement that two numbers or values are equal, such as 6 x 4 = 12 x 2. 2. A noun that counts. When two or more components must be taken into account together in order to comprehend or describe the whole situation, this is known as an equation.

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For a line perpendicular to y = - 5 x - 2 and passing through the point ( 10 , 5 ), the equation in slope-intercept form is y = (1/5)x + 3.

To write an equation for a line perpendicular to y = -5x - 2 and passing through the point (10, 5), we first need to find the slope of the new line. The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. The slope of the original line is -5, so the slope of the new line is 1/5.

Next, we can use the point-slope form of an equation to write the equation for the new line. The point-slope form is y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line. Plugging in the values we have, we get:

y - 5 = (1/5)(x - 10)

Finally, we can rearrange this equation to put it in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. To do this, we'll distribute the 1/5 and then add 5 to both sides of the equation:

y - 5 = (1/5)x - 2
y = (1/5)x + 3

So the equation for the new line is y = (1/5)x + 3. This is our final answer.

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Answer:

2.40 per pound

Step-by-step explanation:

hope that's right

The distance of the ladder from the side of the house can be found using the trigonometric function cosine.

Cosine relates the angle of elevation and the adjacent side (the distance from the side of the house) to the hypotenuse (the length of the ladder).

Cos(55 degrees) = adjacent/hypotenuse

Cos(55 degrees) = adjacent/16

Adjacent = 16 * cos(55 degrees)

Adjacent = 9.2 feet

Therefore, the distance of the ladder from the side of the house is 9.2 feet to the nearest tenth.

Note: To find the cosine of 55 degrees, you can use a calculator or a trigonometric table. The explicit multiplication used in the calculation is 16 * cos(55 degrees).

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A system of inequalities to represent the constraints of this problem are x ≥ 0 and y ≥ 0.

A graph of the system of inequalities is shown on the coordinate plane below.

In order to write a system of linear inequalities to describe this situation, we would assign variables to the number of HD Big View television produced in one day and number of Mega Tele box television produced in one day respectively, and then translate the word problem into algebraic equation as follows:

Since the HD Big View television takes 2 person-hours to make and the Mega TeleBox takes 3 person-hours to make, a linear equation to describe this situation is given by:

2x + 3y = 192.

Additionally, TVs4U’s total manufacturing capacity is 72 televisions per day;

x + y = 72

For the constraints, we have the following system of linear inequalities:

x ≥ 0.

y ≥ 0.

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The probability of exactly two heads in tossing three fair coins is 3/8.

To find the probability of exactly two heads, we need to count the number of outcomes in the sample space that have exactly two heads and divide that by the total number of outcomes in the sample space.
In the sample space {hhh, hht, hth, htt, thh, tht, tth, ttt}, there are three outcomes that have exactly two heads: hht, hth, and thh.
So the probability of exactly two heads is 3/8.

In mathematical terms, this can be represented as:
P(exactly two heads) = number of outcomes with exactly two heads / total number of outcomes
P(exactly two heads) = 3 / 8

Therefore, the probability of exactly two heads in tossing three fair coins is 3/8.

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The radian measure of an angle coterminal to (15π)/4 is (7π)/4. The correct answer is A

To find an angle that is coterminal with (15π)/4 between 0 and 2π, we can subtract or add any multiple of 2π until we get an angle between 0 and 2π.

First, we can simplify (15π)/4 as follows:

(15π)/4 = (4π + π)/4 = π + (π/4)

Now, we can subtract 2π until we get an angle between 0 and 2π:

π + (π/4) - 2π = -π/4

Since -π/4 is negative, we can add 2π to get an angle between 0 and 2π:

-π/4 + 2π = (8π - π)/4 = (7π)/4

Therefore, an angle coterminal with (15π)/4 between 0 and 2π is (7π)/4. The correct answer is A

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The only value of k that satisfies the given inequality and solution set is:

k = 0.

Inequality is defined as mathematical statements that have a minimum of two terms containing variables or numbers that are not equal.

We are given the inequality:

1/2 < k - 4x < 2k + 8

We are also given the solution set of the inequality:

{-3 < x < 2} and {-3 < x < 7/8}

We can simplify the inequality by adding 4x to all parts of the inequality:

1/2 + 4x < k < 2k + 8 + 4x

Next, we can use the given solution set to find the value of k that satisfies the inequality.

If x is between -3 and 2, then the largest possible value of 4x is 8, and the smallest possible value is -12. Therefore:

1/2 + 8 < k < 2k + 8 + 8

or

8.5 < k < 2k + 16

If x is between -3 and 7/8, then the largest possible value of 4x is 7/2, and the smallest possible value is -12. We have:

1/2 + 7/2 < k < 2k + 8 + 7/2

or

4 < k < 2k + 23/2

The intersection of these two solution sets is:

8.5 < k < 2k + 16

and

4 < k < 2k + 23/2

Simplifying the second inequality:

4 < k < 4 + 2k + 23/2 - 2k

or

4 < k < 23/2

Therefore, k must be between 8.5 and 23/2, inclusive.

However, we also need to check if k = 0 satisfies the inequality.

1/2 < 0 - 4x < 2(0) + 8

or

1/2 < -4x < 8

or

-1/8 > x > -2

The solution set {-1/8 > x > -2} is not the same as the given solution set, so k = 0 does not satisfy the inequality.

Therefore, the only value of k that satisfies the given inequality and solution set is:

k = 0.

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The expense E for the production of an item when the price of $74.99 has been established is $3,378,021.86.

To determine the expense E for the production of an item, we need to use the demand function and the given information about fixed expenses and cost per unit.

First, let's plug in the given price of $74.99 into the demand function to find the quantity demanded:

a = 67(74.99) + 74,000

a = 5,024.33 + 74,000

a = 79,024.33

Now that we know the quantity demanded, we can use this information to calculate the total cost of production. The total cost of production is the sum of the fixed expenses and the variable expenses (cost per unit multiplied by quantity demanded):

E = 59,000 + (42.00)(79,024.33)

E = 59,000 + 3,319,021.86

E = 3,378,021.86

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The mathematical expression for the given phrase using x as the variable is:

9x + (x+1)

We can simplify this expression by combining like terms:

9x + x + 1 = 10x + 1

Therefore, the expression is 10x + 1.

Answer: The answer is 21.1304347826 or 21.13 rounded

Step-by-step explanation:

hope this helps

Answer: $30

Step-by-step explanation:

Let x be the cost of one pen
Then x + 1 will be the cost of one book

From the problem, we know that:

5x + 4(x + 1) = 31 (Lester paid $31 for 5 pens and 4 books. A book costs $1.00 more than a pen)

Simplifying the equation:

5x + 4x + 4 = 31

9x = 27

x = 3

So one pen costs $3 and one book costs $4.

Now we can find the cost for Stephan:

6 pens cost 6 x $3 = $18

3 books cost 3 x $4 = $12

So Stephan will pay $18 + $12 = $30.

1.

y= -5x + 4

slope intercept form is y = mx + b (m) being the slope and (b) being the y intercept

The perimeter of the regular octagon with an apothem of 5 units will be 33.14 units.

Since, We know that;

All the sides of the regular polygon are congruent to each other. The perimeter of the regular polygon of n sides will be the product of the number of the side and the side length of the regular polygon.

P = (Side length) x n

Here, The Apothem of a regular octagon is 5 units.

Then the side length of the regular octagon is given as,

tan (360° / (2 × 8)) = (n/2) ÷ 5

tan 22.5° = n / 10

n = 4.142

Then, the perimeter is given as,

P = 8 x 4.142

P = 33.14 units

Thus, The perimeter of the regular octagon with an apothem of 5 units will be 33.14 units.

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The volume of the cone that will fill the cone halfway is [tex]198.72 cm3[/tex] by the given base radius of 9/2cm and 14cm deep.

The  formula for the volume of the cone is

[tex]V = (1/3)πr^2h[/tex]

where h is the height of the cone, r is its base's radius and is a mathematical constant roughly equal to 3.14159.

We must first determine the height of the cone that is filled halfway in to calculate the volume of the cone that will fill it halfway.

Half of the cone's entire height, or 14 cm divided by 2, or 7 cm, corresponds to the height of the half-filled cone.

The cone's base has a radius of 9/2 cm.

We can calculate the volume of the cone that will fill the cone halfway using the formula for a cone's volume as follows:

[tex]V = (1/3)πr^2h[/tex]

[tex]V = (1/3)π(9/2)^2(7) (7)[/tex]

[tex]V ≈ 198.72 cm^3[/tex] (rounded to 2 decimal places) (rounded to 2 decimal places)

Therefore, [tex]198.72 cm3[/tex] is the volume of the cone that will fill it halfway.

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[tex]\sqrt{26^{2} - 24^{2}} = \sqrt{(26-24)(26+24)} = \sqrt{2(50)} = \sqrt{100} = 10 \\[/tex]

[tex]x = \sqrt{10^{2} - 6^{2}} = \sqrt{100 -36} = \sqrt{64} = 8 \\[/tex]

[tex]\implies \bf x = 8[/tex]

Answer:

8

Step-by-step explanation:

Begin with the bigger right angle triangle

Hypotenuse is 26, second side is 24, third side is unknown (a)

26² - 24² = a²

676 - 576 =a²

100 = a²

Therefore a is 10

Now use the value of a which is 10 to solve the smaller triangle

Hypotenuse is 10, one side is 6, third side x is unknown

10²- 6²= x²

100 - 36 =x²

64 =x²

X is 8

The matrix equation for Exercise 57 is:

\[ \begin{bmatrix} 3 & 2 \\ 7 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} -1 \\ 2 \end{bmatrix} \]

In Exercise 57, we are asked to write the given system of linear equations as a matrix equation.

To do this, we need to separate the coefficients of the variables and the constants from the equations and write them in matrix form. The coefficients of the variables will form the coefficient matrix, and the constants will form the constant matrix.

The coefficient matrix will be a 2x2 matrix, with the first row containing the coefficients of x and y from the first equation, and the second row containing the coefficients of x and y from the second equation. The constant matrix will be a 2x1 matrix, with the first row containing the constant from the first equation, and the second row containing the constant from the second equation.

So, the matrix equation for the given system of linear equations will be:

\[ \begin{bmatrix} 3 & 2 \\ 7 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} -1 \\ 2 \end{bmatrix} \]

Therefore, the matrix equation for Exercise 57 is:

\[ \begin{bmatrix} 3 & 2 \\ 7 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} -1 \\ 2 \end{bmatrix} \]

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The given transformations are vertical stretch by a factor of 4, translation 5 units right, and reflection over the x-axis. We can apply these transformations to a function f(x) in the following way:

1. Vertical stretch by a factor of 4: This transformation will multiply the y-values of the function by 4. The new function will be g(x) = 4f(x).

2. Translation 5 units right: This transformation will shift the graph of the function 5 units to the right. The new function will be h(x) = g(x - 5) = 4f(x - 5).

3. Reflection over the x-axis: This transformation will reflect the graph of the function over the x-axis. The new function will be k(x) = -h(x) = -4f(x - 5).

Therefore, the final transformed function will be k(x) = -4f(x - 5).

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The ratios 15 to 18 to 24 in their simplest forms is 5:6:8.

To simplify the ratios 15:18:24, we need to divide all three numbers by their greatest common factor (GCF).

Find the GCF of 15, 18, and 24:

The factors of 15 are 1, 3, 5, and 15.

The factors of 18 are 1, 2, 3, 6, 9, and 18.

The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

The largest factor that all three numbers share is 3, so the GCF is 3.

Divide all three numbers by 3:

15 ÷ 3 = 5

18 ÷ 3 = 6

24 ÷ 3 = 8

Write the simplified ratio:

The simplified ratio is 5:6:8.

Therefore, the simplified ratio of 15:18:24 is 5:6:8.

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The radius of the circle is 13 units

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

The circle

Where we have

Center = (0, 0)

Point - (-5, 12)

The radius of the circle is the distance between the point and the center

So, we have

Radius = √[(0 + 5)² + (0 - 12)²]

Evaluate

Radius = 13

Hence, the radius is 13 units

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Answer:

x = - 2, y = - 8

Step-by-step explanation:

13x - 6y = 22 → (1)

x = y + 6 → (2)

substitute x = y + 6 into (1)

13(y + 6) - 6y = 22

13y + 78 - 6y = 22

7y + 78 = 22 ( subtract 78 from both sides )

7y = - 56 ( divide both sides by 7 )

y = - 8

substitute y = - 8 into (2)

x = y + 6 = - 8 + 6 = - 2

then x = - 2 and y = - 8

Consequently, IS has a 24 length as its length. SR is 44, QS is 55, and SU is 6 shorter than IS.

A physical term known as length pertains to the measurement of an object's length or the separation of two points. Ordinarily, it is expressed in measures like metres, feet, inches, centimetres, etc. One of the basic elements of geometry is length, which is applied in many mathematical and scientific contexts.

given

TU and QR are parallel in the illustration below. Find the length of IS if SU is 6 shorter than IS, QS is 55, and SR is 44.

Since TU and QR are parallel, we have:

angle Angle = ISU QSR (matching angles) (corresponding angles)

angle Angle = SUI SRQ (alternate interior views) (alternate interior angles)

Triangles ISU and QSR are therefore comparable (by angle-angle similarity). So, here we are:

SU/QS Equals IS/SR

Inputting the numbers provided yields:

IS/44 = (IS - 6)/55

The result of multiplying both parts by 44*55 is:

55IS = 44(IS - 6) (IS - 6)

If we simplify, we get:

11IS = 264

IS = 24

Consequently, IS has a 24 length as its length. SR is 44, QS is 55, and SU is 6 shorter than IS.

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The solution to the equation is w=-2. The additive property of equality with integers states that if a=b, then a+c=b+c for any integer c.

This property allows us to add the same number to both sides of an equation in order to isolate the variable and solve for it. In the equation w-4=-6, we can use the additive property of equality to add 4 to both sides in order to isolate the variable w:

w-4+4=-6+4

Simplifying gives us:

w=-2

Therefore, the solution to the equation is w=-2.

In HTML format, the answer would be:

The additive property of equality with integers states that if a=b, then a+c=b+c for any integer c. This property allows us to add the same number to both sides of an equation in order to isolate the variable and solve for it.

In the equation w-4=-6, we can use the additive property of equality to add 4 to both sides in order to isolate the variable w:

w-4+4=-6+4

Simplifying gives us:

w=-2

Therefore, the solution to the equation is w=-2.

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