Sam needs 2/5 pound of turkey to make one sandwich he is going to make 7 sandwiches how many pounds of turkey does he need

The transformation appears to be a rigid motion because A. Yes, because the angle measures and the distances between the vertices are the same as the corresponding angle measures and distances in the preimage.

A rigid motion transformation, colloquially referred to as an isometry, preserves the conformation and magnitude of a geometric construct. This change consists of translations, rotations, and reflections.

For this particular example, the preimage happens to be a right triangle facing leftward, whilst the image is an inverted right triangle facing eastward. This transmutation can be realized through a conjunction of reflection and rotation while maintaining similar angle measurements and distances between vertices. As a result, it is evidently a rigid motion.

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Answer:

  a) multiply by cos²/cos², move sin/cos inside parentheses, simplify

  d) multiply by (cot+cos); use cot=cos·csc, csc²-1=cot² in the denominator

Step-by-step explanation:

You want to prove the identities ...

Usually, we want to prove a trig identity by providing the steps that transforms one side of the identity to the expression on the other side. Here, each of these identity expressions can be simplified, so it is actually much easier to simplify both expressions to one that is common.

We are going to use s=sin(x), c=cos(x), (s/c) = tan(x), and (c/s) = cot(x) to reduce the amount of writing we have to do.

  [tex]s^2\left(\dfrac{c}{s}+1\right)^2=c^2\left(\dfrac{s}{c}+1\right)^2\qquad\text{given}\\\\\\\dfrac{s^2(c+s)^2}{s^2}=\dfrac{c^2(s+c)^2}{c^2}\qquad\text{use common denominator}\\\\\\(c+s)^2=(c+s)^2\qquad\text{cancel common factors; Q.E.D.}[/tex]

Using the same substitutions as above, we have ...

  [tex]\dfrac{c(c/s)}{(c/s)-c}=\dfrac{(c/s)+c}{c(c/s)}\qquad\text{given}\\\\\\\dfrac{c^2}{c(1-s)}=\dfrac{c(1+s)}{c^2}\qquad\text{multiply num, den by s}\\\\\\\dfrac{c(1+s)}{(1-s)(1+s)}=\dfrac{c(1+s)}{c^2}\\\\\\\dfrac{c(1+s)}{1-s^2}=\dfrac{c(1+s)}{c^2}\\\\\\\dfrac{c(1+s)}{c^2}=\dfrac{c(1+s)}{c^2}\qquad\text{Q.E.D.}[/tex]

__

Additional comment

The key transformation in (d) is multiplying numerator and denominator by (1+sin(x)). You can probably prove the identity just by doing that on the left side, then rearranging the result to make it look like the right side.

For (a), the key transformation seems to be multiplying by cos²(x)/cos²(x) and rearranging.

Sometimes it seems to take several tries before the simplest method of getting from here to there becomes apparent. The transformations described in the top "Answer" section may be simpler than those shown in the "Step-by-step" section.

Without a specific set of scatterplots to examine, I can provide some general guidelines for labeling scatterplots based on their association:

1. No association: When there is no pattern or relationship between the two variables being plotted, we label the scatterplot as having no association.

2. Linear positive association: When the points in the scatterplot form a roughly straight line that slopes upwards from left to right, we label the scatterplot as having a linear positive association. This means that as the value of one variable increases, the value of the other variable also tends to increase.

3. Linear negative association: When the points in the scatterplot form a roughly straight line that slopes downwards from left to right, we label the scatterplot as having a linear negative association. This means that as the value of one variable increases, the value of the other variable tends to decrease.

4. Nonlinear association: When the points in the scatterplot do not form a straight line, we label the scatterplot as having a nonlinear association. This means that the relationship between the two variables is more complex and cannot be described simply as a straight line. There are many different types of nonlinear relationships, including curves, U-shaped or inverted-U-shaped patterns, and more.

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There are 715 five-digit positive integers where the digits are non-increasing from left to right.

Here, we have to find the number of five-digit positive integers where the digits are non-increasing from left to right, you can think of this as selecting five digits (from 0 to 9) with repetition allowed, while ensuring that the selected digits are arranged in a non-increasing order.

This is essentially a combinations with repetition problem.

For each digit, there are 10 choices (0 to 9). Since repetition is allowed, you can use a stars and bars approach, where you place 4 bars among 10 possible positions (one for each digit choice) to separate the digits into groups.

The number of ways to arrange 5 digits with repetition allowed is given by the formula:

Number of arrangements = (n + k - 1) choose k,

where n is the number of digits (10 choices) and k is the number of bars (4). Plugging in the values:

Number of arrangements = (10 + 4 - 1) choose 4 = 13 choose 4 = 715.

So, there are 715 five-digit positive integers where the digits are non-increasing from left to right.

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Answer: 3

Step-by-step explanation:

Answer:

2hrs 24 mins

Step-by-step explanation:

Ok so let's make this problem a bit simpler by splitting it up.

Tim paints a room in 6 hours.

So, we can also say that she paints 1/6 of that room in 1 hour

Bella paints it in 4 hours

So, we can also say that she paints 1/4 of that room in 1 hour

Now, lets see what we have:

Bella: 1/4 every hour

Tim: 1/6 every hour

The problem states that they are working together, so we need to add the values we have:

1/4 + 1/6

We cannot just add them, we must make them have the same common denominator.

LCD is 12, you can find that by just doing the times tables for 4 and 6 and seeing what number they match on first.

3/12 + 2/12 = 5/12

So, tim and bella working together paint 5/12 of a room in 1 hour.

They paint 5/12 of a room in 60 minutes

They paint 1/12 of the room in 12 minutes(divide both values by 5)

So if they paint 1/12 of the room in 12 minutes, we can multiply both values by 12 to get our answer.

They paint the full room in 144 minutes(12*12).

144 minutes is 2 hours and 24 minutes

A t-test with a level of significance of 0.05 results in a test statistic of -2.09, indicating a significant difference between the population mean for the math scores and the population mean for the EBRW scores.

To test for a difference between the population mean for the math scores and the population mean for the ebrw scores, we can conduct a two-sample t-test.

Using a calculator or software, we can find that the sample mean for math scores is 520.5 and the sample mean for ebrw scores is 485.5.

The sample size is n = 12 for both groups.

The sample standard deviation for math scores is s1 = 48.50 and for ebrw scores is s2 = 87.63.

Using a level of significance of 0.05, and assuming unequal variances, we can find the test statistic as:

t = (520.5 - 485.5) / sqrt(([tex]48.50^2/12[/tex]) + ([tex]87.63^2/12[/tex]))

t = 0.851

Rounding to two decimal places, the test statistic is 0.85.

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The rate at which the current is changing is -1/32 amperes per second (A/s).

To find the rate at which the current is changing, we will use the given information and apply the differentiation rules. The terms we will use in the answer are voltage (V), resistance (R), current (I), and rate of change.

Given the formula for current: I = V/R
We have V = 10 volts (constant) and dR/dt = 0.2 ohms/second.

We need to find dI/dt, the rate at which the current is changing. To do this, we differentiate the formula for current with respect to time (t):

[tex]dI/dt = d(V/R)/dt[/tex]
Since V is constant, its derivative with respect to time is 0.

dI/dt = -(V * dR/dt) / R^2 (using the chain rule for differentiation)

Now, substitute the given values:

[tex]dI/dt = -(10 * 0.2) / 8^2[/tex]
[tex]dI/dt = -2 / 64[/tex]
[tex]dI/dt = -1/32 A/s[/tex]

The rate at which the current is changing is -1/32 amperes per second (A/s).

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The correct statement is: "The line whose equation is 3x-5y=4 is dilated by a scale factor of [tex]y= (\frac{5}{3} )x[/tex] centered at the origin, and the equation of the dilated line is y= (\frac{5}{3} )x

When a line is dilated by a scale factor of k centered at the origin, the equation of the dilated line is given by y = kx, if the original line passes through the origin. If the original line does not pass through the origin, then the equation of the dilated line is obtained by finding the intersection point of the original line with the line passing through the origin and the point of intersection of the original line with the x-axis, dilating this intersection point by the scale factor k, and then finding the equation of the line passing through this dilated point and the origin.

In this case, the equation of the original line is 3x - 5y = 4. To find the intersection point of this line with the x-axis, we set y = 0 and solve for x:

3x - 5(0) = 4
3x = 4
[tex]x = \frac{4}{3}[/tex]

Therefore, the intersection point of the original line with the x-axis is (4/3, 0). Dilating this point by a scale factor of 5/3 centered at the origin, we obtain the dilated point:

[tex](\frac{5}{3} ) (\frac{4}{3},0) = (\frac{20}{9},0)[/tex]

The equation of the dilated line passing through this point and the origin is given by [tex]y= (\frac{5}{3} )x[/tex]. Therefore, the correct statement is: "The line whose equation is 3x-5y=4 is dilated by a scale factor of [tex]\frac{5}{3}[/tex] centered at the origin, and the equation of the dilated line is [tex]y= (\frac{5}{3} )x[/tex]."

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Number of cars that coasted more than 10.75 feet = 1

To solve this problem, you need to compare the distance each car coasted to 10.75 feet, which is the threshold for determining whether a car coasted more or less than 10.75 feet.

Manuel's car coasted 10 feet, which is less than 10.75 feet, so it did not coast more than 10.75 feet.

Richard's car coasted 10.5 feet, which is also less than 10.75 feet, so it did not coast more than 10.75 feet either.

Diego's car coasted 102 feet, which is more than 10.75 feet. Therefore, only one car coasted more than 10.75 feet, and the answer is 1.

So the answer is:

Number of cars that coasted more than 10.75 feet = 1

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To find the derivative of the function h(p) = -5/(p^2+1), we will use the quotient rule:

h'(p) = [(-5)'(p^2+1) - (-5)(p^2+1)'] / (p^2+1)^2

Simplifying this expression, we get:

h'(p) = (10p) / (p^2+1)^2

To find the values of p such that h'(p) = 0, we will set the numerator equal to 0 and solve for p:

10p = 0

p = 0

Therefore, h'(p) = 0 when p = 0.

To find the values of p in the domain of h such that h'(p) does not exist, we need to find the values of p where the denominator of h'(p) becomes 0:

p^2+1 = 0

This equation has no real solutions, so there are no values of p in the domain of h such that h'(p) does not exist. Therefore, we enter DE (does not exist).

To find the critical numbers of the function, we need to find the values of p where h'(p) = 0 or h'(p) does not exist. We have already found that h'(p) = 0 when p = 0, and we have determined that h'(p) does not exist for any values of p in the domain of h. Therefore, the only critical number of the function is p = 0.
Let's first find the derivative of the given function, h(p) = (p - 5)/(p^2 + 1).

Using the quotient rule, h'(p) = [(p^2 + 1)(1) - (p - 5)(2p)]/((p^2 + 1)^2).

Simplifying, h'(p) = (p^2 + 1 - 2p^2 + 10p)/((p^2 + 1)^2) = (-p^2 + 10p + 1)/((p^2 + 1)^2).

To find the values of p such that h'(p) = 0, set the numerator of h'(p) equal to zero:

-p^2 + 10p + 1 = 0.

This is a quadratic equation, but it does not have any real solutions. Therefore, there are no values of p for which h'(p) = 0, so the answer is DNE.

To find the values of p where h'(p) does not exist, we look for where the denominator is zero:

(p^2 + 1)^2 = 0.

However, this equation has no real solutions, as (p^2 + 1) is always positive. Therefore, there are no values of p for which h'(p) does not exist, so the answer is DE.

Since there are no values of p for which h'(p) = 0 and no values of p for which h'(p) does not exist, there are no critical numbers of the function. The answer is DNE.

Your answer:
h(p) = (p - 5)/(p^2 + 1)
h'(p) = (-p^2 + 10p + 1)/((p^2 + 1)^2)
p (h'(p) = 0) = DNE
p (h'(p) does not exist) = DE
Critical numbers = DNE

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A total number of 59.6001 tiles (approximately 60 tiles) are needed to complete the tiling for the kitchen and bathroom.

To calculate the total area that needs to be tiled, we'll start by converting the given dimensions from millimeters to meters:

Bathroom Inner Dimensions:

Length = 6030 mm = 6.03 m

Width = 5130 mm = 5.13 m

Bathroom Outer Dimensions (including bedroom areas):

Length = 12330 mm = 12.33 m

Width = 4680 mm = 4.68 m

Kitchen Dimensions:

Length = 6030 mm = 6.03 m

Width = 5130 mm = 5.13 m

Total area to be tiled in the bathroom (excluding the bath area):

Area = Length x Width = 6.03 m x (5.13 m - 0.86 m) = 6.03 m x 4.27 m = 25.7701 m²

Total area to be tiled in the kitchen:

Area = Length x Width = 6.03 m x 5.13 m = 30.9919 m²

Total area to be tiled (bathroom + kitchen):

Total Area = 25.7701 m² + 30.9919 m² = 56.762 m²

To account for breakages, we need to purchase 5% more tiles. So, the total number of tiles needed is:

Total Number of Tiles = Total Area x 1.05 (to account for 5% extra)

Total Number of Tiles = 56.762 m² x 1.05 = 59.6001 tiles

The building manager stated that 150 tiles are needed. Comparing this with our calculation:

150 tiles < 59.6001 tiles

Therefore, the statement made by the building manager is not valid. According to our calculations, a total of 59.6001 tiles (approximately 60 tiles) are needed to complete the tiling for the kitchen and bathroom.

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Answer:

B. The graph has a vertical asymptote at

x = -2.

The statement about the graph of the given rational function that is true is: B. The graph has a vertical asymptote at x = -2.

To understand the graph of the rational function f(x) = (3x - 7) / (x + 2), we need to consider its behavior at various points. First, let's investigate the possibility of asymptotes. Asymptotes are lines that the graph approaches but never touches. There are two types of asymptotes: vertical and horizontal.

A vertical asymptote occurs when the denominator of the rational function becomes zero. In this case, the denominator is (x + 2), so we need to find the value of x that makes it zero. Setting x + 2 = 0 and solving for x, we get x = -2. Therefore, the rational function has a vertical asymptote at x = -2 (option B).

To determine if there is a horizontal asymptote, we need to compare the degrees of the numerator and the denominator. The degree of a term is the highest power of x in that term. In the given rational function, the degree of the numerator is 1 (3x) and the degree of the denominator is also 1 (x). When the degrees are the same, we look at the ratio of the leading coefficients, which are 3 (numerator) and 1 (denominator). The ratio of the leading coefficients is 3/1 = 3.

If the ratio of the leading coefficients is a finite value (not zero or infinity), then the rational function will have a horizontal asymptote. In this case, the horizontal asymptote is y = 3 (option C).

Hence the correct option is (b).

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If using the method of completing the square to solve the quadratic equation number 1 be added to both side of the equation to be added to "complete the square".

An algebraic equation of the second degree in x is a quadratic equation. The quadratic equation is written as ax² + bx + c = 0, where x is the variable, a and b are the coefficients, and c is the constant term. The requirement that the coefficient of x² be a non-zero term (a 0) is necessary for an equation to qualify as a quadratic equation. The x² term is written first when constructing a quadratic equation in standard form, then the x term, and finally the constant term.

Add 1 to both sides of the equation to get:

[tex]x^2+4x+4=1[/tex]

The left hand side is now a perfect square:

[tex]x^2+4x+4=(x+2)^2[/tex]

So we have:

[tex](x+2)^2=1[/tex]

Hence:

[tex]x+2=\pm\sqrt{1} =\pm1[/tex]

Subtract 2 from both ends to get:

x = -2 ± 1

That is:

x = -3 or x = -1.

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To create a function that combines the two scenarios, we need to add the amount of money you earn from babysitting to the amount of money you have in your suitcase. We can represent this with the following function:

f(x) = s(x) + a(x)

Where f(x) represents the total amount of money you have after x hours of babysitting. We substitute s(x) with the given function, s(x) = 325, and a(x) with the given function, a(x) = 5(x-2):

f(x) = 325 + 5(x-2)

Simplifying this expression, we can distribute the 5 to get:

f(x) = 325 + 5x - 10

And then combine the constant terms:

f(x) = 315 + 5x

So the function that combines the two scenarios is f(x) = 315 + 5x. This function gives you the total amount of money you will have after x hours of babysitting and taking into account the initial amount of money you have in your suitcase.

In summary, to create a function that combines the two scenarios, we simply add the amount of money earned from babysitting to the initial amount of money in the suitcase. The function f(x) = 315 + 5x represents this total amount of money.

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We have found a value of b that makes y(t) = [tex]e^2t[/tex] [1; 4; -1/2] a solution of y′ = [4 1 3; 2 3 3; −2 −1 −1]y. To check if y(t) is a solution of y′ = Ay, we need to substitute it into the differential equation and see if it holds.

Let's start by finding y′:

y′(t) = [[tex]2e^2t, 8e^2t, 4be^2t[/tex]]

Now, let's find Ay:

Ay = [4 1 3; 2 3 3; −2 −1 −1] [1; 4; b] = [4+4b; 14; -5-b]

We want y(t) = e^2t [1; 4; b] to satisfy y′ = Ay, so we set them equal:

y′ = Ay

[[tex]2e^2t; 8e^2t; 4be^2t] = [4+4b; 14; -5-b] e^2t[/tex] [1; 4; b]

Expanding this equation, we get:

[tex]2e^2t[/tex]= (4+4b)[tex]e^2t[/tex]

[tex]8e^2t[/tex] = 14 [tex]e^2t[/tex]

[tex]4be^2t[/tex]= (-5-b) [tex]e^2t[/tex]

The second equation is always true, so we can ignore it. For the first equation, we can cancel out [tex]e^2t[/tex] on both sides to get:

2 = 4+4b

Solving for b, we get:

b = -1/2

Finally, we can substitute b = -1/2 back into the third equation to check if it holds:

4be^2t = (-5-b) [tex]e^2t[/tex]

-2e^2t = (-5 + 1/2)[tex]e^2t[/tex]

This equation is true, so we have found a value of b that makes y(t) = [tex]e^2t[/tex] [1; 4; -1/2] a solution of y′ = [4 1 3; 2 3 3; −2 −1 −1]y.

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Answer:

2/5 < 0.42 < 43% < 3/7

Step-by-step explanation:

Let's convert them all to decimals:

43% = 0.43

2/5 = 0.4

3/7 = 0.428571...

0.42 = 0.42

Now we can arrange them in ascending order:

0.4

0.42

0.43

0.428571...

Answer:

Step-by-step explanation:

625

Answer:

Step-by-step explanation:

The total balance of the two investment accounts at the end of ten years will be $16,564.08. To calculate the total balance of the two accounts at the end of ten years,

we need to use the formulas for simple interest and compound interest.

For Account I, the simple interest formula is:

I = Prt

where I is the interest earned, P is the principal (the amount deposited), r is the annual interest rate as a decimal, and t is the time in years.

Plugging in the values for Account I, we get:

I = (3500)(0.065)(10) = $2,275

So, after ten years, the balance in Account I will be:

B1 = P + I = 3500 + 2275 = $5,775

For Account II, the compound interest formula is:

A = P(1 + r/n)^(nt)

where A is the balance at the end of the time period, P is the principal, r is the annual interest rate as a decimal, n is the number of times the interest is compounded per year, and t is the time in years.

Plugging in the values for Account II, we get:

A = 5900(1 + 0.06/1)^(1*10) = $10,789.08

So, after ten years, the balance in Account II will be $10,789.08.

Therefore, the total balance of the two accounts at the end of ten years will be:

Total balance = Balance in Account I + Balance in Account II

= $5,775 + $10,789.08

= $16,564.08

In summary, by using the formulas for simple interest and compound interest, we can calculate that the total balance of the two investment accounts at the end of ten years will be $16,564.08.

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In circle P, if m(QR) = 80 , and m(QRT) = 39 , m(QPR) = 39 and m(PT) = 78

Based on the information given, we know that:

- m(QR) = 80 (this is the measure of arc QR)
- m(QRT) = 39 (this is the measure of angle QRT)

To find the other measures, we can use the following formulas:

- The measure of a central angle is equal to the measure of its intercepted arc
- The measure of an inscribed angle is half the measure of its intercepted arc

Using these formulas, we can find the measure of angle QPR and the measure of arc PT as follows:

- m(QPR) = m(QRT) = 39 (since angle QRT and angle QPR intercept the same arc QR)
- m(PT) = 2 * m(QRT) = 78 (since angle QRT and angle PQT intercept the same arc PT, and the measure of an inscribed angle is half the measure of its intercepted arc)

So the final answers are:

- m(QR) = 80
- m(QRT) = 39
- m(QPR) = 39
- m(PT) = 78

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The surface area of the storage bin is 34.8ft²². Storage capacity will the storage bin have 13.5ft³.

The area of a square is composed of (Side) (Side) square units. The area of a square equals d22 square units when the diagonal, d, is known. For instance, a square with sides that are each 8 feet long is 8 8 or 64 square feet in area. (ft2).

a=2*5-2>.5

b = 2 .

c = 2.1 ft

S = 2(3 * 2 * 2) + 3 * 2 + 2 * 1/v * 1/v

x 2+ 1 2 *3+3*1*3

= 2 deg + 6 + 1 + 1.5 + 6 * 3

= 34.8ft²

V = V Triangle prism + Vandrangular prism.

= 3×2×2 + 2 x = x2x2

= 12+ 1.5

= 13.5ft³

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Question:

Dave wants to know the amount of material he needs

to buy to make the bin. What is the surface area of the

storage bin?

Part B

How much storage capacity will the storage bin have?

The inflation rate for Ken's gas expenses between the two years is approximately 42.26%.

To calculate the inflation rate for Ken's gas expenses, we can use the following formula: (Current Year Expense - Previous Year Expense) / Previous Year Expense × 100%.

In this case, the previous year's gas expense was $1,098 and the current year's expense is $1,562.

To find the difference in expenses, subtract the previous year's expense from the current year's expense: $1,562 - $1,098 = $464.

Now, divide this difference by the previous year's expense: $464 / $1,098 ≈ 0.4226.

Finally, multiply the result by 100% to get the inflation rate: 0.4226 × 100% ≈ 42.26%.

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Step-by-step explanation:

[tex] \frac{1}{( {x}^{2} + 1) } = \frac{u}{v} [/tex]

u = 1

u' = 0

v = x² + 1

v' = 2x

[tex] \frac{1}{ ({x}^{2} + 1)} \\ = \frac{u'v - v'u}{ {v}^{2} } \\ = \frac{0 - (2x \times 1)}{ {( {x}^{2} + 1)}^{2} } \\ = - \frac{2x}{ { ({x}^{2} + 1) }^{2} } [/tex]

the MAXIMUM area that can be enclosed is 900 m²

To find the maximum area that can be enclosed, we need to find the vertex of the parabolic function A(x) = -4x^2 + 120x. The vertex represents the maximum point on the parabola.

The x-coordinate of the vertex can be found using the formula x = -b/2a, where a is the coefficient of the x^2 term and b is the coefficient of the x term. In this case, a = -4 and b = 120, so x = -120/(2*(-4)) = 15.

To find the y-coordinate of the vertex, we can substitute x = 15 into the function: A(15) = -4(15)^2 + 120(15) = 900. Therefore, the maximum area that can be enclosed is 900 square meters.

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Answer: C

Step-by-step explanation:

C is a parallelogram, meaning that both sets of opposite sides are parallel.

Answer:  9

Step-by-step explanation:

The angle is 1/2 of the arc angle

Since the tangent line is a line, I know the angle on the other side of 78 is

180-78 = 102

That angle, 102, is 1/2 the arc angle

102  = 1/2 (23x -3)                 > multiply both sides by 2

204 =  23x -3                        > add 3 to both sides

207 = 23x                             >divide both sides by 23

x=9

For the expressions  3x+4 –2x, and 5x+2x+x the simplified expression after combining like terms is x+4 and 8x.

The given expressions are  3x+4 –2x, and 5x+2x+x

We have to simplify these expressions by combining the like terms

For the expression 3x+4 –2x

We have to combine like terms

x+4

Now for expression  5x+2x+x

Combine the like terms to get

8x

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The best measure of central tendency for the data set below { 10, 18, 13, 11, 62, 12, 17, 15} is option B- Mean because there is no outlier.

The best measure of central tendency for the given data set depends on the nature of the data and what you want to represent.

If you want to find the middle value of the data set that is not affected by the outlier, then the median is the best measure of central tendency. In this case, the median is 13, as it is the middle value when the data is arranged in ascending order.

If you want to find the typical or average value of the data set, then the mean is the best measure of central tendency. In this case, the mean is approximately 20, calculated by adding all the values and dividing by the total number of values.

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You need the unit price $0.40/mile to find the cost of a 47-mile taxi ride.

Use the unit price to find the cost of a 47-mile taxi ride

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Answer:

2

Step-by-step explanation:

The K and N are the trapeziums and the two lines opposite to them go in a parallel line

The given statement "If ∑an is a convergent series, then S = limsn, where sn is the nth partial sum. " is true. This is because the sum of the series is defined as the limit of the sequence of partial sums.

Given that ∑an is a convergent series, sn is the nth partial sum, S=limsn

To prove limn→∞ an = 0

Since ∑an is convergent, we know that the sequence {an} must be a null sequence, i.e., it converges to 0. This means that for any ε>0, there exists an N such that |an|<ε for all n≥N.

Now, let's consider the partial sums sn. We know that S=limsn, which means that for any ε>0, there exists an N such that |sn−S|<ε for all n≥N.

Using the triangle inequality, we can write:

|an|=|sn−sn−1|≤|sn−S|+|sn−1−S|<2ε

Therefore, we have shown that limn→∞ |an| = 0, which implies limn→∞ an = 0, as required.

Hence, the proof is complete.

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