exact volume of this cylinder ​

94.2 square feet is the surface area of the cylinder

The given cylinder has a height of 2 ft and radius of 3 ft

We have to find the surface area of cylinder

Surface area =2πrh+2πr²

Substitute the values of radius and height

=2πr(r+h)

=2×3.14×3(3+2)

=18.84(5)

=18.84×5

Surface area=94.2 square feet

Hence, the surface area of the cylinder is 94.2 square feet

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The perimeter of the dilated rectangle M'N'O'P' is 54 centimeters.

If the area of the original rectangle is A, and the area of the dilated rectangle is A', then the relationship between their areas is:

A' = k² × A

Similarly, the relationship between their perimeters is:

P' = k × P

The original rectangle has an area of 14 square centimeters.

The dilated rectangle has an area of 126 square centimeters.

We need to find the perimeter of the dilated rectangle.

Let's denote the scale factor as "k."

126 = k² × 14

Dividing both sides by 14, we get:

9 =  k²

k = 3

Now, we can find the perimeter of the dilated rectangle using the perimeter relationship:

P' = k × P

The original rectangle has a perimeter of 18 centimeters, so:

P' = 3 × 18

P' = 54 centimeters

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Applying linear functions, we have: a. Bakery B charges a higher delivery fee    b. Bakery A charges a higher price per cookie

c. Bakery B is cheaper.

In order to solve this problem, we will need to write the linear function for bakery A and bakery B.

Where:

b = the initial delivery fee which is the y-intercept

m = price per cookie

x = number of cookies

y = total cost

For Bakery A:

b = 5 (initial delivery fee)

m = price per cookie = change in y/change in x = 10 - 5 / 2 - 0

m = 5/2 = $2.5.

The linear function for bakery A is: y = 2.5x + 5.

For Bakery B:

b = 7 (initial delivery fee)

m = price per cookie = change in y/change in x = 16 - 7 / 4 - 0

m = 9/4 = $2.25.

The linear function for bakery A is: y = 2.25x + 7.

Therefore, we have:

a. The bakery that charges a higher delivery fee is Bakery B.

b. The bakery that charges a higher price per cookie is Bakery A.

c. Substitute x = 12 into both linear functions:

Bakery A: y = 2.5(12) + 5 = $35

Bakery B: y = 2.25(12) + 7 = $34

The bakery that is cheaper is: Bakery B.

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The values of the angles a, b and c are 60°, 110° and 80° respectively.

Given is a figure we need to find the measure of the angles given,

a, b, c.

So,

70° + 50° + a = 180° [making linear pair angles]

120° + a = 180°

a = 60°

50° + a = b [corresponding angles between parallel lines]

b = 50° + 60°

b = 110°

a + 40° + ∠HFE = 180° [angle sum property of a triangle]

60° + 40° + ∠HFE = 180°

∠HFE = 80°

∠HFE = c [alternate angles between parallel lines]

c = 80°

Hence the values of the angles a, b and c are 60°, 110° and 80° respectively.

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The statement "x + 3 is a factor of p(x) = [tex]x^3 - 7x^2 + 15x - 9[/tex]" is false.

To determine whether x + 3 is a factor of p(x) = [tex]x^3 - 7x^2 + 15x - 9[/tex], we can use the factor theorem.

According to the factor theorem, if a polynomial p(x) has a factor (x - a), then p(a) = 0.

In this case, we have x + 3 as a possible factor. To check if it is indeed a factor, we substitute -3 for x in the polynomial p(x):

[tex]p(-3) = (-3)^3 - 7(-3)^2 + 15(-3) - 9[/tex]

= -27 - 63 - 45 - 9

= -144

Since p(-3) is not equal to zero, we conclude that x + 3 is not a factor of p(x).

Therefore, the statement "x + 3 is a factor of [tex]p(x) = x^3 - 7x^2 + 15x - 9[/tex]" is false.

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Answer:

[tex]-61[/tex]

Step-by-step explanation:

[tex]\mathrm{We\ have\ f(}x)=-3x^2-2x+4\\\mathrm{Put}\ x=-5,\\\mathrm{f}(-5)=-3(-5)^2-2(-5)+4=-3(25)+10+4=-75+10+4=-61[/tex]

i) Compute the autonomous investment, government the MPM is 0.4.

AI = 1 / (1 - 0.35) = 1 / 0.65 ≈ 1.538 (approximately)

GM = 1 / (1 - 0.35) = 1 / 0.65 ≈ 1.538 (approximately)

XM = 1 / (1 - 0.4) = 1 / 0.6 ≈ 1.667 (approximately)

TM = -0.35 / (1 - 0.35) ≈ -0.538 (approximately)

ii) The equilibrium income is approximately 709.52.

To compute the autonomous investment (A), government expenditure (G), export (X), and tax (T) multipliers, we need to use the following formulas:

Autonomous Investment Multiplier (AI):

AI = 1 / (1 - MPC)

where MPC is the marginal propensity to consume.

Government Expenditure Multiplier (GM):

GM = 1 / (1 - MPC)

Export Multiplier (XM):

XM = 1 / (1 - MPM)

where MPX is the marginal propensity to import.

Tax Multiplier (TM):

TM = -MPC / (1 - MPC)

Now let's calculate these multipliers using the given information:

i). Compute the multipliers:

MPC = change in consumption / change in income

Here, we can see that the consumption function is given by C = 40 + 0.35Y.

So, the MPC is 0.35.

MPM = change in imports / change in income

Here, we can see that the import function is given by M = 45 + 0.4Y.

So, the MPM is 0.4.

AI = 1 / (1 - 0.35) = 1 / 0.65 ≈ 1.538 (approximately)

GM = 1 / (1 - 0.35) = 1 / 0.65 ≈ 1.538 (approximately)

XM = 1 / (1 - 0.4) = 1 / 0.6 ≈ 1.667 (approximately)

TM = -0.35 / (1 - 0.35) ≈ -0.538 (approximately)

ii). Find the equilibrium income:

To find the equilibrium income, we set aggregate demand (Y) equal to aggregate supply (Y) and solve for Y.

Aggregate demand (Y):

Y = C + I + G + (X - M)

Substituting the given values, we have:

Y = (40 + 0.35Y) + (70 - 2r) + 400 + (240 - (45 + 0.4Y))

Simplifying the equation:

Y = 40 + 0.35Y + 70 - 2(0.5) + 400 + 240 - 45 - 0.4Y

Combining like terms:

Y = 745 - 0.05Y

Bringing Y terms to one side:

1.05Y = 745

Dividing both sides by 1.05:

Y = 745 / 1.05 ≈ 709.52 (approximately)

Therefore, the equilibrium income is approximately 709.52.

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The 10,100 cards will be necessary to build a 100-story card tower.

To build a 100-story card tower, we can use the triangular number formula to calculate the total number of cards needed. The formula is:

Triangular number = (n * (n + 1)) / 2

In this case, n represents the number of stories in the tower (100). Plug in the value for n:

Triangular number = (100 * (100 + 1)) / 2
Triangular number = (100 * 101) / 2
Triangular number = 10,100 / 2
Triangular number = 5,050

Additionally, each story requires two cards, so we multiply the triangular number by 2 to get the total number of cards:

Total cards = 5,050 * 2
Total cards = 10,100

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The radius of the cone with the given volume which is is terms of π would be = 18m

To determine the radius of the cone, the formula that should be used in the formula for the volume of cone which would be given below;

Volume of cone =1/3πr²h

where;

radius = ?

volume = 5,400π

height = 50m

That is;

5,400π = 1/3×π×r²×50

make r² the subject of formula;

r² = 5400π×3/50π

= 108×3

= 324

r = √324

= 18

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The function f(x) = 1/x³ is indeed a decreasing function for x > 0.

To show that the function f(x) = 1/x³ is a decreasing function, we need to demonstrate that the derivative of f(x) is negative for all x > 0.

Let's calculate the derivative of f(x) with respect to x:

f'(x) = d/dx (1/x³)

Using the power rule for differentiation, we can rewrite the expression as:

[tex]f'(x) = (-3) / x^{(3+1)[/tex]

Simplifying further, we have:

f'(x) = -3 / x⁴

Now, to show that f(x) is decreasing, we need to prove that f'(x) < 0 for all x > 0.

Substituting x = 1 in f'(x), we get:

f'(1) = -3 / 1⁴ = -3

Since -3 is negative, we can conclude that f'(x) < 0 for all x > 0.

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Answer: 109.875

Step-by-step explanation: 879 / 8 = 109.875 because if you divide 8 by 9 you get 1 so after you would divide 79 by 8 = 9.875 so then you put 109 then after but the decimals 109.875.

Answer:

X

Step-by-step explanation:

Reflective symmetry, also known as mirror symmetry, exists when a figure can be reflected across a line and appear the same. Rotational symmetry exists when a figure can be rotated about a point by a certain degree less than 360 and still appear the same.

Among the uppercase letters W, X, Y, and Z:

The letter W has no reflective or rotational symmetry.

The letter X has both reflective and rotational symmetry. It's symmetrical on both the vertical and horizontal axes and has a rotational symmetry of 180 degrees.

The letter Y only has reflective symmetry on the vertical axis, but it does not have rotational symmetry.

The letter Z also has reflective symmetry on the horizontal axis, but it does not have rotational symmetry.

So among these letters, only X has both reflective and rotational symmetry.

The value of a is given as follows:

a = 2.

The rate of change is given as follows:

2.4 inches per hour.

The slope-intercept equation for a linear function is presented as follows:

y = mx + b

In which:

The line passes through the points (–1, –5) and (4, 5), hence the slope m is obtained as follows:

m = (5 - (-5))/(4 - (-1))

m = 2.

Hence:

y = 2x + b.

When x = -1, y = -5, hence the intercept b is obtained as follows:

-5 = -2 + b

b = -3.

Hence:

y = 2x - 3.

Then the value of a is obtained as the value of x when y = 1, hence:

2x - 3 = 1

2x = 4

x = 2.

The rate of change in the snowfall is given as follows:

m = (8.1 - 2.3)/2

m = 2.4 inches per hour.

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Answer: Tommy's sister is four meters away from him.

Step-by-step explanation:

If Tommy's older brother is 4 meters ahead and he is 5 meters from his sister,subtract 1 from that since it is described she is right next to him,and you get four.

This is my first time answering anyone so I hope it helps.

Answer:

y=3.5x

Step-by-step explanation:

to find the the feet mary walks per second, divide 14 by 4

14/4=3.5 feet/second

the total distance mary walks can be represented by multiplying the number of feet she walks per second by the number of seconds she walks

y= 3.5x

in this model, x is seconds

We have that Linda's net and expression incorrect.

Now, With the Expression of the surface area of attic

45 (40 + 25 + 25) + 1/2 (40 x 15)

Hence, Surface area is,

X = 4050 + 600

X = 4650 feet²

Hence, The answer is No, Because Her expression for the triangles are incorrect;

1/2(40 x 15)

Instead of,

1/2 x 40 x 15

Hence, Linda's net and expression incorrect,

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Complete question is,

Linda is adding padding to all of the surfaces inside her attic for extra warmth in the winter.

She needs to find the approximate surface area of the attic, including the walls, floor, and ceiling. The attic is in the shape of a triangular prism. Linda draws the net and writes the expression below to represent the surface area of the attic. Are Linda's net and expression correct?

Answer:

The equation for those inputs and outputs is:
0.25 *input - 151.75 = Output  

Step-by-step explanation:

For this, first search for the Slope of the funtion
THe slope can be found by calculating this :
(y2-y1)/(x2-x1)  or in this case (output2 - output1)/ (input2 - input1)

IT continues as follows:
( -48.5 - 24)  / (-413 - (-703) ) =
= (-72.5) / (290)
= -0.25

The slope is -0.25. If the function is a line then the equation is

-0.25*input + B = output

with any of the choices we can determine B.
  -0.25 * (-703) + B    = 24

   175.75 + B              = 24
175.75 +B - 175.75    = 24 - 175.75

               B                 = - 151.75

The equation for those inputs and outputs is:
0.25 *input - 151.75 = Output  

8.1 CO: oxide

8.2 CO₂: dioxide

8.3 SO₂: dioxide

8.4 CaCO:

8.5 NaOH:

8.1 CO: oxide

CO represents carbon monoxide, which is composed of one carbon atom (C) and one oxygen atom (O). It is classified as an oxide because it consists of oxygen combined with another element.

8.2 CO₂: dioxide

CO₂ represents carbon dioxide, which consists of one carbon atom (C) and two oxygen atoms (O). It is classified as a dioxide because it contains two oxygen atoms per molecule.

8.3 SO₂: dioxide

SO₂ represents sulfur dioxide, composed of one sulfur atom (S) and two oxygen atoms (O). It is also classified as a dioxide because it contains two oxygen atoms per molecule.

8.4 CaCO: This formula is incomplete or incorrect. It seems to be missing a subscript or superscript to indicate the number of atoms or ions involved. Please provide the correct formula, and I'll be happy to match it with its type.

8.5 NaOH: hydroxide

NaOH represents sodium hydroxide, which consists of one sodium ion (Na⁺) and one hydroxide ion (OH⁻). It is classified as a hydroxide because it contains the hydroxide ion, which is composed of one oxygen atom and one hydrogen atom.

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The value of the home after 5 years would be $194,415.

How to calculate the value of the home after 5 years

First, let's calculate the increase in value for each 3-year period:

After the first 3 years:

Increase = 9% of $150,000 = 0.09 * $150,000 = $13,500

After the second 3 years:

Increase = 9% of ($150,000 + $13,500) = 0.09 * ($150,000 + $13,500) = $14,850

After the third 3 years:

Increase = 9% of ($150,000 + $13,500 + $14,850) = 0.09 * ($150,000 + $13,500 + $14,850) = $16,065

Now, let's add these increases to the initial value of the home to find the value after 5 years:

Value after 5 years = $150,000 + $13,500 + $14,850 + $16,065 = $194,415

Therefore, the value of the home after 5 years would be $194,415.

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Answer:

4. 4155.3cm³

5. 783m³

6. 1962.44821094mm³

#4 Explanation:

First, you would separate the 2 shapes it consists of, a cylinder and a cone.

For the cone, you would use the equation (1/3)πr²h. Fill in the appropriate variables with the numbers from the formula. Pi, π, is a number itself so you leave it as it is.

r = radius = 8cm

h = height = 14cm

(1/3)(π)(8)²(14)

Put this equation into a calculator, it's important to include the parentheses or it might give you the incorrect answer, this is so that it separates the different values while multiplying. After inputting it, it should give you:

v = 938.289005872cm³

Then, you would have to find the volume of the cylinder, using the equation; πr²h.

The radius would be the same as the cone, and the height is as the problem states, 16cm.

Solve the problem after inputting the values:

π(8)²(16)= 3216.99087728cm³

Then, add the values to get the total volume.

938.289005872cm³ + 3216.99087728cm³ =

4155.27988315cm³

Round to nearest tenth: 4155.3cm³

#5 Explanation:

Again, you would separate the shapes into 2 separate ones.

The top one is a triangular prism and uses the equation (1/2)bh.

b = base = 9m

h = 12m

(1/2)(9)(12) = 54m³

The bottom shape is a rectangular prism, where we use the equation L×W×H.

Length = 9m

Width = 9m

Height = 9m

Substitute the numbers into the equation.

9×9×9 = 729m³

Add both of them together to get the total volume:

54m³ + 729m³ = 783m³

#6 Equation:
In this figure, we are looking for the outer shape, so we will have to subtract the smaller cone.

First, we find the total volume, and we will use the same equation, (1/3)πr²h.

radius = half of diameter = 20/2 = 10mm

h = 20mm

(1/3)π(10)²(20) = 2094.39510239mm³

Second, we find the volume of the smaller cone, using the same equation, (1/3)πr²h.

In order to find the diameter, we will have to subtract the 7 mm that are on either side of the cone, so 20 - 14 = 6, then divide it by 2 so we can get the necessary radius. 6/2 = 3

r = 3mm

h = 14mm

Substitute values into equation and use calculator to solve.

(1/3)π(3)²(14) = 131.946891451mm³

Subtract the value of the smaller cone from total volume in order to get the volume for this figure.

2094.39510239mm³ - 131.946891451mm³ =

1962.44821094mm³

Answer:

5 m

Step-by-step explanation:

To convert 500 cm to m, you simply divide by 100 since there are 100 centimeters in 1 meter.

So, 500 cm ÷ 100 = 5 m

Answer:

  100

Step-by-step explanation:

You want the value of --(5-1)(5²).

The even number of leading minus signs cancel each other, so the effect is as if there were no leading minus signs.

  --(5 -1)(5²)

  = --(4)(25)

  = --(100)

  = -(-100)

  = 100

__

Additional comment

As always, the value of a numeric expression can be found using a calculator.

<95141404393>

The solutions are : solution of the following system of equations are:

(- 1/2, 1/2)

(- 1, 2)

Here, we have,

The given system of equations is expressed as

y = 2x²- - - - - - - - - - - - - - -1

y = - 3x - 1- - - - - - - - - -- - - -2

We would apply the method of substitution by substituting equation 1 into equation 2. It becomes

2x² = - 3x - 1

2x² + 3x + 1 = 0

We would find two numbers such that their sum or difference is 3x and their product is 2x².

The two numbers are 2x and x. Therefore,

2x² + 2x + x + 1 = 0

2x(x + 1) + 1(x + 1) = 0

2x + 1 = 0 or x + 1 = 0

x = - 1/2 or x = - 1

Substituting x = - 1/2 into equation 1, it becomes

y = 2(-1/2)² = 1/2

Substituting x = - 1 into equation 1, it becomes

y = 2(-1)² = 2

Hence, The solutions are : solution of the following system of equations are:

(- 1/2, 1/2)

(- 1, 2)

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The expression that is a factor of both x² - 9 and x² + 8x + 15 is (x + 3).

Given the polynomials in the question:

x² - 9

x² + 8x + 15

To find a factor that is common to both x² - 9 and x² + 8x + 15, we can factorize the two expressions and look for any common factors.

First, factorize x² - 9:

Using the difference of sqaure formula

a² - b² = ( a + b )( a - b )

x² - 9

x² - 3² = (x - 3)(x + 3)

Next, factorize x² + 8x + 15:

Using AC method

( x + 3 )( x + 5 )

From the factorizations, we can see that both expressions have a common factor of (x + 3).

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1) The value of x = 12°

2) For any real value of n, y = n and  x = (3/2)n

And the measure of angle B is 68°

3) The measure of angle A is 26° and the measure of angle B is 154°

4) The coordinates of vertex D(4, 0)

1)

We know that a linear pair of angles always add up to 180°

168° + x° = 180°

x = 180° - 168°

x = 12°

2)

Here, in parallelogram ABCD, the measure of angle A = 112°

We know that the opposite angles of a parallelogram are equal.

So, m∠C = 112°

Let measure of angle B and D is p degrees.

∠A + ∠B + ∠C + ∠D = 360°

2p + 2(112) = 360°

p = (360 - 224)/2

p = 68°

so, the measure of angle B is 68°

The opposite sides of a parallelogram are equal.

So, side AD = side BC

(2x + 4) = (3y + 4)

2x = 3y

x = (3/2)y

For any real value of n,

y = n

then x = (3/2)n

3)

Consider parallelogram ABCD.

Let us assume that one of the interior angle measures x and then other interior angle be y.

x is 50 degrees more than 4 times the measure of y

x = 4y + 50

We know that the sum of two interior angles (non-opposite) is 180°

⇒ x + y =  180°

⇒ 4y + 50 + y =  180°

⇒ 5y = 130°

⇒ y = 26°

And the value of x would be,

x = 4(26) + 50

x = 154°

So, the measure of angle A is 26° and the measure of angle B is 154°

4)

Let the fourth vertex be D(a,b)

Since ABCD is a parallelogram, the diagonals bisect each other.

coordinates of midpoint of AC = coordinates of midpoint of BD

By midpoint formula,

((1 + 6)/2, (2 + 4)/2) = ((3 + a)/2 , (6 + b)/2)

comparing x and y coordinates,

(3 + a)/2 = (1 + 6)/2

3 + a = 7

a = 4

(6 + b)/2 = (2 + 4)/2

6 + b = 6

b = 0

So, the coordinates of vertex D(4, 0)

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The percentage of buyers who paid between 150,000 and 153,300 is approximately 68% + 2.5% = 70.5%.

To solve this problem, we can use the properties of the normal distribution and the empirical rule (also known as the 68-95-99.7 rule) to estimate the percentage of buyers who paid between 150,000 and 153,300.

According to the empirical rule, given a normal distribution:

approximately 68% of the data falls within one standard deviation of the mean approximately 95% of the  three standard deviations of the mean, the data are contained.

In this case, we want to find the percentage of buyers who paid between 150,000 and 153,300, which is one interval of length 3300 above the mean. To use the empirical rule, we need to standardize this interval by subtracting the mean and dividing by the standard deviation:

z1 = (150,000 - 150,000) / 1100 = 0

z2 = (153,300 - 150,000) / 1100 = 3

Here, z1 represents the number of standard deviations between 150,000 and the mean, and z2 represents the number of standard deviations between 153,300 and the mean.

Since the interval we are interested in is within three standard deviations of the mean (z2 <= 3), we can use the empirical rule to estimate the percentage of buyers who paid between 150,000 and 153,300:

Approximately 68% of the buyers paid within one standard deviation of the mean, which is between 149,000 and 151,000 (using z-scores of -1 and 1).

Approximately 95% of the buyers paid within two standard deviations of the mean, which is between 148,000 and 152,000 (using z-scores of -2 and 2).

Therefore, the remaining percentage of buyers who paid between 152,000 and 153,300 is approximately (100% - 95%) / 2 = 2.5%.

So, the percentage of buyers who paid between 150,000 and 153,300 is approximately 68% + 2.5% = 70.5%.

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There are no solutions to the system of inequalities Option (d)

Inequalities are a fundamental concept in mathematics and are commonly used in solving problems that involve ranges of values.

A system of two inequalities is a set of two inequalities that are considered together. In this case, the system of inequalities is

y > 3x + 1

y < 3x - 3

The inequality y > 3x + 1 represents a line on the coordinate plane with a slope of 3 and a y-intercept of 1. The inequality y < 3x - 3 represents another line on the coordinate plane with a slope of 3 and a y-intercept of -3. We can draw these lines on the coordinate plane and shade the regions that satisfy each inequality.

The first line has a positive slope and goes through (negative 2, negative 5) and (0, 1). Everything to the left of the line is shaded. The second line has a positive slope and goes through (0, negative 3) and (1, 0). Everything to the right of the line is shaded.

We can start by analyzing the inequality y > 3x + 1. This inequality represents the region above the line with a slope of 3 and a y-intercept of 1. Therefore, any point that is above this line satisfies this inequality.

Next, we analyze the inequality y < 3x - 3. This inequality represents the region below the line with a slope of 3 and a y-intercept of -3. Therefore, any point that is below this line satisfies this inequality.

To determine which values satisfy both inequalities, we need to find the region that satisfies both inequalities. This region is the intersection of the regions that satisfy each inequality.

When we analyze the regions that satisfy each inequality, we see that there is no region that satisfies both inequalities. Therefore, there are no values that satisfy the system of inequalities shown.

There are no solutions to the system of inequalities y > 3x + 1 and y < 3x - 3 by analyzing the regions that satisfy each inequality on a coordinate plane. The lack of a solution is determined by the fact that there is no region that satisfies both inequalities.

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Complete Question :

Which is true about the solution to the system of inequalities shown?

y > 3x + 1

y < 3x – 3

On a coordinate plane, 2 solid straight lines are shown. The first line has a positive slope and goes through (negative 2, negative 5) and (0, 1). Everything to the left of the line is shaded. The second line has a positive slope and goes through (0, negative 3) and (1, 0). Everything to the right of the line is shaded.

Options:

a)Only values that satisfy y > 3x + 1 are solutions.

b)Only values that satisfy y < 3x – 3 are solutions.

c)Values that satisfy either y > 3x + 1 or y < 3x – 3 are solutions.

d)There are no solutions.

Answer:

D

Step-by-step explanation:

The standard notation for the given scientific notation[tex]5.397 * 10^2[/tex] is 539.7

To convert the given scientific notation [tex]5.397 * 10^2[/tex]to standard notation, you simply need to multiply the coefficient (5.397) by the power of [tex]10 (10^2).[/tex]

The power of 10 indicates how many places the decimal point should be moved to the right. In this case, [tex]10^2[/tex] means moving the decimal point two places to the right.

Multiplying the coefficient 5.397 by

[tex]10^2[/tex]

gives us:

[tex]5.397 \times 10^2 = 5.397 \times \ 100 = 539.7[/tex]

Therefore, the standard notation for the given scientific notation 5.397 x [tex]10^2[/tex]  is 539.7.

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The critical values t₀ for a two-sample t-test is ± 2.0.6

To find the critical values t₀ for a two-sample t-test to test the claim that the population means are equal (i.e., µ₁ = µ₂), we need to use the following formula:

t₀ = ± t_(α/2, df)

where t_(α/2, df) is the critical t-value with α/2 area in the right tail and df degrees of freedom.

The degrees of freedom are calculated as:

df = (s₁²/n₁ + s₂²/n₂)² / [(s₁²/n₁)²/(n₁-1) + (s₂²/n₂)²/(n₂-1)]

n₁ = 14, n₂ = 12, X₁ = 6,X₂ = 7, s₁ = 2.5 and s₂ = 2.8

α = 0.05 (two-tailed)

First, we need to calculate the degrees of freedom:

df = (s₁²/n₁ + s₂²/n₂)² / [(s₁²/n₁)²/(n₁-1) + (s₂²/n₂)²/(n₂-1)]

= (2.5²/14 + 2.8²/12)² / [(2.5²/14)²/13 + (2.8²/12)²/11]

= 24.27

Since this is a two-tailed test with α = 0.05, we need to find the t-value with an area of 0.025 in each tail and df = 24.27.

From a t-distribution table, we find:

t_(0.025, 24.27) = 2.0639 (rounded to four decimal places)

Finally, we can calculate the critical values t₀:

t₀ = ± t_(α/2, df) = ± 2.0639

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