Trapezium: Parallel side 1 is 7cm. Parallel side 2 is 11cm. Height is 6cm. What will be the area? Please show your working.

Answer:

axis of symmetry=(1.5,6.5)

x intercept= (-1,1.5)

y intercept=(4,0)

State the null and alternative hypotheses, determine the level of significance, Calculate the test statistic also determine the critical value.

Step 1: Express the invalid and elective speculations,

The invalid speculation is that the change of halting distances on wet asphalt is equivalent to or not exactly the fluctuation of halting distances on dry asphalt.

H0: σ2(wet) ≤ σ2(dry)

The other possibility is that the variance of stopping distances on wet pavement is greater than that on dry pavement.

Ha: σ2(wet) > σ2(dry)

Step 2: Choose the appropriate test and determine the significance level,

The level of significance is 5%. Since we are comparing two variances of normally distributed populations, we will use the F-test.

Step 3: Calculate the test statistic,

The F-test measurement is determined as the proportion of the example differences:

F = s2(wet)/s2(dry)

where s2(wet) and s2(dry) are the sample variances of stopping distances on wet and dry pavement, respectively.

F = 32²/16² = 4

Step 4: Determine the critical value

The critical value for the F-test with 15 and 15 degrees of freedom (16-1 and 16-1) at a 5% level of significance is 2.54 (from F-tables).

Step 5: Settle on a choice and decipher the outcomes,

Since the calculated F-value of 4 is greater than the critical value of 2.54, we reject the null hypothesis. We can conclude that the variance of stopping distances on wet pavement is greater than the variance of stopping distances on dry pavement.

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Answer:

Interest Earned: $360

Total value of the Account: $1260

Step-by-step explanation:

To find the interest earned, we can use the formula:

Interest = Principal x Rate x Time

where the principal is the amount deposited, the rate is the interest rate expressed as a decimal, and the time is the number of years the money is left in the account.

In this case, the principal is $900, the rate is 0.04 (since 4% is equivalent to 0.04 as a decimal), and the time is 25 years. Plugging in these values, we get:

Interest = $900 x 0.04 x 25 = $900 x 1 = $360

So the interest earned over 25 years is $360.

To find the total value of the account, we can add the interest earned to the original deposit:

Total Value = Principal + Interest

Total Value = $900 + $360 = $1260

So the total value of the account after 25 years is $1260.

Hope this helps!

An equation that would describe the line of best fit is

Using the line of best fit, a prediction of Spot's weight after 18 months is 35 pounds.

In order to determine a linear equation for the line of best fit (trend line) that models the data points contained in the graph, we would use the point-slope equation:

y - y₁ = m(x - x₁)

Where:

First of all, we would determine the slope of this line;

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope (m) = (20 - 5)/(10 - 2)

Slope (m) = 15/8

Slope (m) = 1.875

At data point (2, 5) and a slope of 1.875, a linear equation for the line of best fit can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - 5 = 1.875(x - 2)  

y = 1.875x + 1.25

When x = 18, the weight is given by:

y = 1.875(18) + 1.25

y = 35 pounds.

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Complete Question:

John recorded the weight of his dog Spot at different ages as shown in the scatter plot below.

Part A:

Write an equation that would describe the line of best fit.

Part B:

Using the line of best fit, make a prediction of Spot's weight after 18 months.

The relationship between j and k can be expressed as j = k^(-3) * C, where C is a constant. To find the value of C, we can use the initial condition j = 3 when k = 6:

3 = 6^(-3) * C

C = 3 * 6^3 = 648

So the relationship is j = 648 / k^3. To find another possible value for j and k, we can simply plug in a different value for k:

For option A:

j = 648 / 2^3 = 81

For option B:

j = 648 / 3^3 = 24

For option C:

j = 648 / 2^3 = 81

For option D:

j = 648 / 81^3 = 0.0008

For option E:

j = 648 / 2^3 = 81

Therefore, the only option that is another possible value for j and k is (A) j = 18, k = 2.

The sd2 is 19.76 ( not listed ).

To find sd2 (the standard deviation squared) for the data set x1 = 18, x2 = 10, x3 = 7, x4 = 5, and x5 = 11, follow these steps:
1. Calculate the mean: (18 + 10 + 7 + 5 + 11) / 5 = 51 / 5 = 10.2
2. Calculate the squared deviations from the mean: (18 - 10.2)^2 = 60.84, (10 - 10.2)^2 = 0.04, (7 - 10.2)^2 = 10.24, (5 - 10.2)^2 = 27.04, (11 - 10.2)^2 = 0.64
3. Calculate the average of squared deviations: (60.84 + 0.04 + 10.24 + 27.04 + 0.64) / 5 = 98.8 / 5 = 19.76

The sd2 (standard deviation squared) for the given data set is 19.76, which is not listed among the given options (a. 15.1, b. 18.3, c. 20.2, d. 24.7).

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To find the distance Add the distance between each point and the y-axis, the correct option is D.

We are given that;

Two points  A(-8,-4) and B(2,-4)

Now,

To find the distance between two points on a coordinate plane, you can use the distance formula: d = √((x2 - x1)² + (y2 - y1)²), where (x1, y1) and (x2, y2) are the coordinates of the points. In this case, the points are A(-8, -4) and B(2, -4). Plugging in the values, we get:

d = √((2 - (-8))² + (-4 - (-4))²) d = √(10² + 0²) d = √(100) d = 10

The distance between the points is 10 units.

Therefore, by the distance answer will be Add the distance between each point and the y-axis.

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your answer :- the value of dy/dt is 28/3.

To find dy/dt, we need to take the derivative of both sides of the equation 3xy-4x+6y^3= -108 with respect to t:

d/dt(3xy-4x+6y^3) = d/dt(-108)

Using the product rule and chain rule, we get:

3x(dy/dt) + 3y(dx/dt) - 4(dx/dt) + 18y^2(dy/dt) = 0

Substituting the given values of dx/dt, x, and y, we get:

3(6)(dy/dt) + 3(-2)(-12) - 4(-12) + 18(-2)^2(dy/dt) = 0

Simplifying and solving for dy/dt, we get:

18dy/dt - 24 - 72 - 72 = 0

18dy/dt = 168

dy/dt = 168/18

dy/dt = 28/3

Therefore, the value of dy/dt is 28/3.

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Answer:

True.

In the data set 19, 8, 7, 5, 4, 9, 2, 5, 8, 6, the value 19 is an outlier. An outlier is a data point that is significantly different from the rest of the data points in a set. In this case, the value 19 is much higher than the other values in the set. This could be due to a number of factors, such as a data entry error or a genuine outlier.

There are a number of ways to identify outliers. One common method is to use the interquartile range (IQR). The IQR is the difference between the third and first quartiles of a data set. A data point that is more than 1.5 times the IQR above the third quartile or below the first quartile is considered to be an outlier.

In this case, the value 19 is more than 1.5 times the IQR above the third quartile. Therefore, it is considered to be an outlier.

Outliers can be removed from a data set, or they can be left in. Removing outliers can sometimes improve the accuracy of statistical analysis, but it is important to be careful not to remove too many data points. Leaving outliers in can sometimes make the data set more difficult to analyze, but it can also provide useful information about the data.

Step-by-step explanation:

Answer:

True

Step-by-step explanation:

Outliers are numbers far from the rest of the numbers.

In the image below, Center A is located at the center of the circle and is marked by a red dot.

Circle is a two-dimensional geometric figure that consists of all points in a plane that are at a given distance from a given point, the centre. A circle is a simple closed shape in Euclidean geometry. It is the set of all points in a plane that are at the same distance from its centre. A circle can be defined as the locus of a point moving at a constant distance from a fixed point. It is one of the most basic shapes in geometry. It is also known as a circular arc, a closed curve, and a simple closed curve. Circular shapes are used in many areas of design, from logos to furniture.

Diameter DE is marked with a dashed line and is the longest line that can be drawn within the circle, extending from point D to point E. Radius AC is marked with a solid line, extending from point A to point C. Central angle PAC is marked by the red arc and is the angle between points P and A. Lastly, tangent OB is marked with a dotted line and is a straight line that touches the circle at only one point (point B). This line is perpendicular to the radius AC.

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complete questions as follows-

Draw your own circle and illustrate the following:
1. Center A
2. Diameter DE
3. Radius AC
4. Central angle PAC
5. Tangent OB​

X and Y are not independent.

P(Y > 2X) = 3/16.

Marginal pdf of X = 12x(1-x)² for 0 < x < 1

(a) To determine if X and Y are independent, we need to check if the joint pdf can be factored into the product of the marginal pdfs:

f(x,y) = 24xy if 0 < x, 0 < y, x + y < 1, and zero otherwise.

Marginal pdf of X can be calculated by integrating the joint pdf over the all possible values of y:

f(x) = ∫ f(x,y) dy from 0 to 1-x

= ∫ 24xy dy from 0 to 1-x

= 12x(1-x)² for 0 < x < 1

Similarly, the marginal pdf of Y can be found by integrating the joint pdf over all possible values of x:

f(y) = ∫ f(x,y) dx from 0 to 1-y

= ∫ 24xy dx from 0 to 1-y

= 12y(1-y)² for 0 < y < 1

To check for independence, we need to verify if f(x,y) = f(x)f(y) for all x and y. However, if we multiply the marginal pdfs, we get:

f(x)f(y) = 144xy(1-x)²(1-y)² for 0 < x < 1 and 0 < y < 1

This is not the same as the joint pdf, so X and Y are not independent.

(b) To find P(Y > 2X), we need to integrate the joint pdf over the region where Y > 2X:

P(Y > 2X) = ∫∫ f(x,y) dA over the region where Y > 2X

= ∫∫ 24xy dA over the region where Y > 2X

= ∫∫ 24xy dxdy over the region where 0 < y < 2x and x+y < 1

= ∫[0,1/2] ∫[y/2,1-y] 24xy dxdy

= 3/16

Therefore, P(Y > 2X) = 3/16.

(c) The marginal pdf of X is given by:

f(x) = ∫ f(x,y) dy from 0 to 1-x

= ∫ 24xy dy from 0 to 1-x

= 12x(1-x)² for 0 < x < 1

Same result we get in part (a).

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X and Y are not independent.

P(Y > 2X) = 3/16.

Marginal pdf of X = 12x(1-x)² for 0 < x < 1

(a) To determine if X and Y are independent, we need to check if the joint pdf can be factored into the product of the marginal pdfs:

f(x,y) = 24xy if 0 < x, 0 < y, x + y < 1, and zero otherwise.

Marginal pdf of X can be calculated by integrating the joint pdf over the all possible values of y:

f(x) = ∫ f(x,y) dy from 0 to 1-x

= ∫ 24xy dy from 0 to 1-x

= 12x(1-x)² for 0 < x < 1

Similarly, the marginal pdf of Y can be found by integrating the joint pdf over all possible values of x:

f(y) = ∫ f(x,y) dx from 0 to 1-y

= ∫ 24xy dx from 0 to 1-y

= 12y(1-y)² for 0 < y < 1

To check for independence, we need to verify if f(x,y) = f(x)f(y) for all x and y. However, if we multiply the marginal pdfs, we get:

f(x)f(y) = 144xy(1-x)²(1-y)² for 0 < x < 1 and 0 < y < 1

This is not the same as the joint pdf, so X and Y are not independent.

(b) To find P(Y > 2X), we need to integrate the joint pdf over the region where Y > 2X:

P(Y > 2X) = ∫∫ f(x,y) dA over the region where Y > 2X

= ∫∫ 24xy dA over the region where Y > 2X

= ∫∫ 24xy dxdy over the region where 0 < y < 2x and x+y < 1

= ∫[0,1/2] ∫[y/2,1-y] 24xy dxdy

= 3/16

Therefore, P(Y > 2X) = 3/16.

(c) The marginal pdf of X is given by:

f(x) = ∫ f(x,y) dy from 0 to 1-x

= ∫ 24xy dy from 0 to 1-x

= 12x(1-x)² for 0 < x < 1

Same result we get in part (a).

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Answer:

Para calcular la cantidad de minutos que Carlos entrena cada día, necesitamos sumar el tiempo que pasa en el gimnasio por la mañana y por la tarde, en minutos.

Por la mañana, Carlos entrena desde las 07:20 hasta las 09:55. Para calcular el tiempo en minutos, podemos restar los minutos de inicio (20) de los minutos de final (55) en la hora de inicio (07), y luego multiplicar el resultado por 60 (porque hay 60 minutos en una hora). Así:

(09 - 07) horas x 60 minutos/hora + (55 - 20) minutos = 2 x 60 + 35 minutos = 120 + 35 minutos = 155 minutos

Por la tarde, Carlos entrena desde las 18:05 hasta las 19:40. Podemos hacer el mismo cálculo:

(19 - 18) horas x 60 minutos/hora + (40 - 05) minutos = 1 x 60 + 35 minutos = 60 + 35 minutos = 95 minutos

Entonces, en total, Carlos entrena 155 minutos por la mañana y 95 minutos por la tarde, lo que suma un total de 250 minutos al día.

The solution to the initial-value problem is y(t) = u(t-1)( [tex]e^5^(^t^-^1^)[/tex] - [tex]e^-^4^(^t^-^1^)[/tex]  + sin(3(t-1))), which satisfies both the differential equation and the initial conditions.

Using the Laplace transform, we can solve the initial-value problem y"-9y+20y=u(t-1) with y(0)=0 and y'(0)=1. T

aking the Laplace transform of both sides and simplifying, we get Y(s) = (1/s² + 9s - 20)(1/ [tex]e^s[/tex] ). Using partial fractions, we can write this as Y(s) = (1/(s-5)) - (1/(s+4)) + (1/s² + 9s).

Taking the inverse Laplace transform and using the shift theorem, we get y(t) = u(t-1)( [tex]e^5^(^t^-^1^)[/tex] -  [tex]e^-^4^(^t^-^1^)[/tex]   + sin(3(t-1))). Therefore, the solution to the initial-value problem is y(t) = u(t-1)( [tex]e^5^(^t^-^1^)[/tex] -  [tex]e^-^4^(^t^-^1^)[/tex]   + sin(3(t-1))), where u(t-1) is the unit step function.

The Laplace transform is a powerful tool in solving initial-value problems in differential equations. By taking the Laplace transform of both sides, we can transform the differential equation into an algebraic equation, which is easier to solve.

In this case, we used partial fractions to simplify the Laplace transform and then used the inverse Laplace transform to find the solution to the initial-value problem.

The shift theorem was used to incorporate the initial condition y(0)=0.

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To rephrase the question, we need to find the highest common factor of 72 and 56.

72 = 2^3 x 3^2

56 = 7 x 2^3

So, the highest common factor of 72 and 56 is 2^3, or 8.

Therefore, the longest possible cut of wire Larry can have is 8cm.

Answer:

I don’t know the answer

Step-by-step explanation:

A linear equation that best represent the given data is: A. y = 4.6x + 26.5.

In this scenario, the number of times fertilized would be plotted on the x-axis (x-coordinate) of the scatter plot while the yield of crop per acre would be plotted on the y-axis (y-coordinate) of the scatter plot through the use of Microsoft Excel.

On the Microsoft 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 a linear 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 times fertilized and the yield of crop per acre, a linear equation for the line of best fit is given by:

y = 4.6x + 26.5

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The fraction that represents the amount of chips each student should receive is 1/9 or 2/18. (Option 4)

The problem states that 18 students want to share two bags of chips equally. Therefore, we need to divide the chips into 18 equal parts to find the amount each student should receive. We can represent this as:

2 bags of chips = 18 equal parts

To find the fraction of chips each student should receive, we need to divide the total number of parts (18) by the number of students (18):

18 parts ÷ 18 students = 1 part/student

Therefore, each student should receive 1 part out of the 18 total parts. We can express this as a fraction:

1 part/18 parts = 1/18

Since we have two bags of chips, each containing 1/18 of the total chips, we can add them together to get the total amount of chips each student should receive:

1/18 + 1/18 = 2/18

Simplifying this fraction, we get:

2/18 = 1/9

Therefore, each student should receive 1/9 of the total chips.

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Complete Question:

A class of 18 students wants to share two bags of chips equally. Which fraction represents the amount of chips each student should receive?

By using the method of Lagrange multipliers, minimum and maximum values of function are  -3√17 and 3√17, respectively.

We can use the method of Lagrange multipliers to find the minimum and maximum values of the function subject to the given constraint. The Lagrange function is

L(x, y, z, λ) = 3x + 2y + 4z - λ(x^2 + 2y^2 + 6z^2 - 17)

Taking partial derivatives with respect to x, y, z, and λ, we get

∂L/∂x = 3 - 2λx = 0

∂L/∂y = 2 - 4λy = 0

∂L/∂z = 4 - 12λz = 0

∂L/∂λ = x^2 + 2y^2 + 6z^2 - 17 = 0

Solving these equations, we get:

x = ±√(17/2), y = ±√(17/8), z = ±√(17/24), and λ = 1/17

We evaluate the function at all eight possible combinations of these values and find that the minimum value is -3√17 and the maximum value is 3√17.

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For each given set of information:

1)sin x = -5/13, x in Quadrant III

sin 2x = -0.96, cos 2x = 0.28, tan 2x = -3.42

2)tan x = -1/4, cos x > 0

sin 2x = -0.48, cos 2x = 0.88, tan 2x = -0.55

3)sin x = 5/13, x in Quadrant I

sin 2x = 0.87, cos 2x = 0.48, tan 2x = 1.81

4)sin x = 5/13, csc x < 0

sin 2x = -0.87, cos 2x = 0.48, tan 2x = -1.81

The Double-Angle Formula for Sine is: sin 2x = 2sin x cos x.

The Half-Angle Formula for Sine is: sin(x/2) = ±√[(1 - cos x) / 2].

Since sin x = -5/13 and x is in Quadrant III, we know that cos x is negative. We can use the formula for sin 2x to find sin 2x = 2sin x cos x = 2(-5/13)(-12/13) = -0.96. Similarly, we can find cos 2x = cos²x - sin²x = (12/13)² - (-5/13)² = 0.28, and tan 2x = sin 2x / cos 2x = -3.42.

We know that tan x = -1/4 and cos x > 0. Using the Pythagorean identity, we can find sin x = √(1 - cos²x) = √(1 - (16/17)²) = -5/17 (since x is in Quadrant IV, sin x is negative). Using the formula for sin 2x, we can find sin 2x = 2sin x cos x = 2(-5/17)(16/17) = -0.48.

Similarly, we can find cos 2x = cos²x - sin²x = (16/17)² - (-5/17)² = 0.88, and tan 2x = sin 2x / cos 2x = -0.55.

Since sin x = 5/13 and x is in Quadrant I, we know that cos x is positive. Using the formula for sin 2x, we can find sin 2x = 2sin x cos x = 2(5/13)(12/13) = 0.87. Similarly, we can find cos 2x = cos²x - sin²x = (12/13)² - (5/13)² = 0.48, and tan 2x = sin 2x / cos 2x = 1.81.

Since sin x = 5/13 and csc x < 0, we know that x is in Quadrant IV. Using the formula for sin 2x, we can find sin 2x = 2sin x cos x = 2(5/13)(-12/13) = -0.87. Similarly, we can find cos 2x = cos²x - sin²x

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Option B is showing graph of given equations.

Rewrite the first equation in slope-intercept form, y = mx + b:

y = -3 + (1/2)x

This equation has a y-intercept of -3 and a slope of 1/2.

Rewrite the second equation in slope-intercept form:

3x + 2y = 2

2y = -3x + 2

y = (-3/2)x + 1

This equation has a y-intercept of 1 and a slope of -3/2.

Plot the y-capture of every situation on the y-hub:

The principal condition has a y-block of - 3, so plot the point (0,- 3).

The subsequent condition has a y-capture of 1, so plot the point (0,1).

Find additional points on each line by utilizing the slope of each equation:

Since the slope of the first equation is 1/2, the point (2,-2.5) can be obtained by moving up one unit and right two units from the y-intercept (0,-3). To get the point, move up one unit and right two units again.

Since the slope of the second equation is -3/2, move down three units and right two units from the y-intercept (0,1) to reach the point (2,-3.5). To get the point, move down three units and right two units once more.

Plot the points and draw lines through them:

Plot the points (0,-3), (2,-2.5), and (4,-2) for the first equation, and connect them with a straight line. Plot the points (0,1), (2,-3.5), and (4,-5) for the second equation, and connect them with a straight line.

The resulting graph should show two lines intersecting at a point and  the intersection point of the two lines is (2, -2). This point represents the solution to the system of equations.

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If United Bank offers a 15-year mortgage at an APR of 6.2%.

a. Monthly Payment  is $1,025.90

b. Total Interest is  $64,662

c. Monthly Payment  $810.55

d. Total Interest is $123,165

e. Difference in the monthly payments is $215.35.

f.  You could save 10 years of payments

Using this formula to find the monthly payment

Monthly Payment = P * (r * (1 + r)^n) / ((1 + r)^n - 1)

Total Interest = (Monthly Payment * n) - P

where

P= principal amount borrowed

r = monthly interest rate (APR / 12)

n = total number of monthly payments

a. United Bank Monthly payment

P = $120,000

r = 6.2% / 12 = 0.00517

n = 15 years * 12 months/year = 180

Monthly Payment = 120000 * (0.00517 * (1 + 0.00517)^180) / ((1 + 0.00517)^180 - 1)

Monthly Payment  = $1,025.90

b.  United Ban Total interest

Total Interest = ($1,025.90* 180) - 120000

Total Interest = $64,662

c. Capitol Bank Monthly payment

P = $120,000

r = 6.5% / 12 = 0.00542

n = 25 years * 12 months/year = 300

Monthly Payment = 120000 * (0.00542 * (1 + 0.00542)^300) / ((1 + 0.00542)^300 - 1)

Monthly Payment  = $810.55

d. Capitol Bank Total interest

Total Interest = (810.55 * 300) - 120000

Total Interest = $123,165

e. Capitol Bank has the higher total interest by $58,503 ( $123,165 - $64,662).

f. The difference in the monthly payments is:

$1025.90 - $810.55= $215.35.

g. You could save 10 years of payments if you took up a 15-year mortgage as opposed to a 25-year mortgage.

Therefore the Monthly Payment  is $1,025.90.

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Step-by-step explanation:

To find the area of the rectangular piece, we can use the formula:

Area = length x width

We can find the length of the rectangle by calculating the difference between its two x-coordinates:

Length = |-14 - (-18)| = 4 ft

We can find the width of the rectangle by calculating the difference between its two y-coordinates:

Width = |13 - 5| = 8 ft

Therefore, the area of the rectangular piece is:

Area = 4 ft x 8 ft = 32 ft^2

To find the total area of the garden, we need to add the area of the existing garden (which is given as 20 square feet) to the area of the new rectangular piece:

Total area = 20 ft^2 + 32 ft^2 = 52 ft^2

Therefore, the total area of the garden is 52 square feet. The answer is 52 ft^2.

Answer:

Step-by-step explanation:

You want the areas of two circles, one with radius 4 cm, the other with diameter 12 m.

The area of a circle is given by the formula ...

  A = πr²

The radius (r) is half the diameter, so the second circle's radius is 6 m.

The area is ...

  π(4 cm)² = 16π cm² ≈ 50.3 cm²

The area is ...

  π(6 m)² = 36π m² ≈ 113.1 m²

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Let f be the function defined x^3 for x< or =0 or x for x>o.

The correct answer is (D) f ‘(x)>0 for x not equal 0.

(A)f is an odd function

f is not an odd function because f(-x) does not equal -f(x) for all x.

(B) f is discontinuous at x=0

f is continuous at x=0

because the limit of f as x approaches 0 from the left is 0 and the limit of f as x approaches 0 from the right is also 0, and these limits are equal to f(0)=0.

(C) f has a relative maximum

f does not have a relative maximum because f(x) increases as x increases for x>0 and decreases as x decreases for x<0, but there is no point where f(x) is greater than all nearby values of f.

(D) f ‘(x)>0 for x not equal 0

f ‘(x) = 3x^2 for x<0 and 1 for x>0, which is greater than 0 for all x not equal to 0.

(E) This statement is not true because (D) is true.

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Let f be the function defined x^3 for x< or =0 or x for x>o.

The correct answer is (D) f ‘(x)>0 for x not equal 0.

(A)f is an odd function

f is not an odd function because f(-x) does not equal -f(x) for all x.

(B) f is discontinuous at x=0

f is continuous at x=0

because the limit of f as x approaches 0 from the left is 0 and the limit of f as x approaches 0 from the right is also 0, and these limits are equal to f(0)=0.

(C) f has a relative maximum

f does not have a relative maximum because f(x) increases as x increases for x>0 and decreases as x decreases for x<0, but there is no point where f(x) is greater than all nearby values of f.

(D) f ‘(x)>0 for x not equal 0

f ‘(x) = 3x^2 for x<0 and 1 for x>0, which is greater than 0 for all x not equal to 0.

(E) This statement is not true because (D) is true.

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The countably infinite number of guests can be accommodated by moving each current guest to the next room number (i.e., guest in room 1 moves to room 2, guest in room 2 moves to room 3, and so on) and then assigning the newly arrived guests to the vacated rooms (i.e., guest 1 goes to room 1, guest 2 goes to room 2, and so on).

This way, every guest still has a room and no one is evicted.

The solution to this problem relies on the fact that the set of natural numbers (which is countably infinite) can be put into one-to-one correspondence with the set of positive even numbers (also countably infinite).

This means that each current guest can be moved to the next room number without any gaps or overlaps, and the newly arrived guests can be assigned to the vacated rooms in the same manner.

The concept of one-to-one correspondence is fundamental to understanding countability and infinite sets, and it plays a key role in many mathematical proofs and arguments.

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Yes, if f 0 (x) < 0 for 1 < x < 6, then f is decreasing on (1, 6). This is because a function is decreasing on an interval if its derivative is negative on that interval.

Given that f'(x) < 0 for 1 < x < 6, we can conclude that the function f(x) is decreasing on the interval (1, 6).
Here's a step-by-step explanation:
1. f'(x) represents the first derivative of the function f(x) with respect to x.
2. The first derivative f'(x) gives us information about the slope of the tangent line to the curve of the function f(x) at any point x.
3. If f'(x) < 0 for 1 < x < 6, it means that the slope of the tangent line is negative for every x in the interval (1, 6).
4. A negative slope indicates that the function is decreasing at that interval.
So, given the information provided, we can confirm that the function f(x) is decreasing on the interval (1, 6). Since f 0 (x) is the derivative of f(x), if it is negative for 1 < x < 6, then f(x) must be decreasing on that interval.

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The approximate area under the curve y=x^3 from x=1 to x=4 using a right endpoint approximation with 6 subdivisions is 49.0125 square units.

To approximate the area under the curve y=x^3 from x=1 to x=4 using a right endpoint approximation with 6 subdivisions, we can use the following steps:

1. Determine the width of each subdivision by dividing the total width (4-1) by the number of subdivisions (6):

width = (4-1)/6 = 0.5

2. Determine the right endpoint of each subdivision by adding the width to the left endpoint:

x1 = 1, x2 = 1.5, x3 = 2, x4 = 2.5, x5 = 3, x6 = 3.5

3. Evaluate the function at each right endpoint:

f(x1) = f(1) = 1
f(x2) = f(1.5) = 3.375
f(x3) = f(2) = 8
f(x4) = f(2.5) = 15.625
f(x5) = f(3) = 27
f(x6) = f(3.5) = 42.875

4. Calculate the area of each rectangle by multiplying the width by the function value at the right endpoint:

A1 = 0.5*1 = 0.5
A2 = 0.5*3.375 = 1.6875
A3 = 0.5*8 = 4
A4 = 0.5*15.625 = 7.8125
A5 = 0.5*27 = 13.5
A6 = 0.5*42.875 = 21.4375

5. Add up the areas of all the rectangles to get an approximate area under the curve:

A ≈ A1 + A2 + A3 + A4 + A5 + A6 = 49.0125

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Check the picture below.

[tex]\textit{area of a trapezoid}\\\\ A=\cfrac{h(a+b)}{2}~~ \begin{cases} h~~=height\\ a,b=\stackrel{parallel~sides}{bases~\hfill }\\[-0.5em] \hrulefill\\ a=6\\ b=10\\ h=11 \end{cases}\implies A=\cfrac{11(6+10)}{2}\implies A=88[/tex]

The angle between AFE is measured (A) 42°.

Since ΔABC is an equilateral triangle, all its angles are 60°. Since CAD-18°,: ∠CAE = ∠CAD + ∠DAE = 18° + 60° = 78°.

Since AC is the angle bisector of ∠BCD,:

∠ACB = ∠ACD = (1/2)∠BCD. Since ΔABC is equilateral, ∠BCA = 60°.

Therefore, ∠BCD = ∠BCA + ∠ACB = 60° + (1/2)∠BCD, which implies that ∠ACB = 30°.

Since BE- CD,:

∠BEC = ∠BCD - ∠CED = ∠ACB - ∠CED = 30° - ∠CED.

Since ∠CAF = 12°,:

∠BAC = ∠CAD + ∠DAF = 18° + 12° = 30°.

Therefore, ∠BCA = 30°, and BC = AC.

Let x = ∠CED. Since BE = CD and BC = AC,: CE = AD = BC = AC.

In ΔCED,: ∠ECD = 180° - ∠CED - ∠CDE = 180° - x - 60° = 120° - x.

In ΔCAD,: ∠CAD + ∠CDA + ∠ACD = 180°, which implies that ∠CDA = 60° - (1/2)∠CAD = 60° - 9° = 51°.

In ΔADF,: ∠ADF = 180° - ∠BAC - ∠DAF = 180° - 30° - 12° = 138°.

In ΔAFE,: ∠AFE = ∠ACB + ∠BEC + ∠CED + ∠ECD + ∠CDA + ∠ADF = 30° + (180° - 30° - x) + x + (120° - x) + 51° + 138° = 489° - x.

Since the angles of a triangle sum to 180°:

∠AFE + ∠EAF + ∠AEF = 180°

∠AFE + 60° + 78° = 180°

∠AFE = 42°.

Therefore, the answer is (A) 42°.

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