What is the fourth vertex of the rectangle with vertices at (1,4), (1,−4), and (−5,−4)?
Group of answer choices

(4,4)

(4,-5)

(-5,-5)

(-5,4)

This confirms that our polynomial function has the correct y-intercept.

To find a polynomial function with the given properties, we can use the fact that the zeros of a polynomial function are the values of x that make the function equal to zero. The multiplicity of a zero is the number of times that zero appears as a factor in the polynomial.

Since we are given a triple zero at x = 1, this means that (x - 1)^3 is a factor of the polynomial. Similarly, since we are given a double zero at x = 3, this means that (x - 3)^2 is a factor of the polynomial.

To find the y-intercept, we can set x = 0 and solve for y. Since the y-intercept is given as (0, 27), this means that the constant term of the polynomial is 27.

Putting all of this information together, we can write the polynomial function as:

f(x) = 27(x - 1)^3(x - 3)^2

This is a polynomial function that satisfies the given properties. It has a triple zero at x = 1, a double zero at x = 3, and a y-intercept of (0, 27).

We can check our answer by plugging in the values of x and seeing if the function equals zero:

f(1) = 27(1 - 1)^3(1 - 3)^2 = 0
f(3) = 27(3 - 1)^3(3 - 3)^2 = 0

Both of these values equal zero, which confirms that our polynomial function has the correct zeros. Additionally, when we plug in x = 0, we get:

f(0) = 27(0 - 1)^3(0 - 3)^2 = 27

This confirms that our polynomial function has the correct y-intercept.

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Answer: Your welcome!

Step-by-step explanation:

14

13

12

11

10

9

8

7

6

O

6

7 8 9 10 11 12 13

10

1415

The answer is 14, 13, 12, 11, 10, 9, 8, 7, 6, O, 6, 7, 8, 9, 10, 11, 12, 13, 10, 14, 15. This is the result of translating the shape by the vector (5, 4). The original shape was shifted 5 units to the right and 4 units down, resulting in the new shape.

Answer:

x = 8

Step-by-step explanation:

4(x + 3) = 44 ( divide both sides by 4 )

x + 3 = 11 ( subtract 3 from both sides )

x = 8

Answer:

  76 ft²

Step-by-step explanation:

You want the area of a trapezoid with bases 6 ft and 13 ft, and height 8 ft.

The area formula for a trapezoid is ...

  A = 1/2(b1 +b2)h

  A = 1/2(6 ft +13 ft)(8 ft) = 76 ft²

The area of the figure is 76 square feet.

__

Additional comment

The length of the longer base on the right is the sum of 6 ft and 7 ft. It is 13 ft.

The area of the figure is 76 square feet.

the area of a trapezoid is: 76 ft².

Here, we have,

from the given figure, we get,

The length of the longer base on the right is the sum of 6 ft and 7 ft. It is 13 ft.

We have to find the area of a trapezoid with bases 6 ft and 13 ft, and height 8 ft.

Trapezoid

The area formula for a trapezoid is

 A = 1/2(b₁ +b₂)h

 A = 1/2(6 ft +13 ft)(8 ft)

     = 76 ft²

so, we get,

The area of the figure is 76 square feet.

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No, the result is not significant at 1 percent level for the p-value of 0.802.

Results are significant at a certain level,

Compare the p-value to the significance level (known as alpha).

The significance level is typically set at 0.05 or 0.01.

Here, the p-value is 0.802, which is much larger than 0.01.

This implies here we cannot reject the null hypothesis at a 1% significance level.

Or , the result is not statistically significant at a 1% level.

If the significance level was 5%  that is equal to 0.05.

The result would not be significant at 5% level either.

If the significance level was 10% which is equal to 0.10.

Then the result would be significant as the p-value is less than 0.10.

Therefore, for the given p-value 0.802 result is not significant at 1%.

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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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Answer:

Twenty Seven (27) cubic yards

Answer:

volume =27 cubic yards

Step-by-step explanation:

Volume = Base Area x Height

Volume= 3·3·3

Volume= 27

Answer: 1

Step-by-step explanation:

3(x-2) = 4x+2

3x-6 = 4x+2

To move 3x, deduct it on both sides of the equation.

3x - 6 - (3x) = 4x + 2 - (3x)

0 - 6 = [1]x +2

Given:-

[tex] \: [/tex]

Solution:-

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

━━━━━━━━━━━━━━━━━━━━━━━

hope it helps ⸙

Answer:

78

Step-by-step explanation:

The domain of the piecewise function g of x is all real numbers, except for x = 2, which is the point at which the two functions overlap. Therefore, the domain is (–∞, 2) ∪ (2, ∞).

A domain is a collection of computers and devices connected to one another through a network. It is a logical grouping of computers and devices, such as a local area network (LAN) or a wide area network (WAN). Each device within the network is assigned a unique address, allowing it to communicate with other devices within the network. Domains can also be used to control access to resources, such as files, folders, printers, and databases. Domains can be used to authenticate users and set permissions, which determine the type of access a user has to the domain’s resources. This is commonly used in businesses and organizations to ensure that only authorized personnel have access to sensitive information.

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Answer:

cos G = 3/5

sin G = 4/5

tan G = 4/3

Step-by-step explanation:

I(9, 12)

GH = 9

HI = 12

(GI)² = 9² + 12²

GI = 15

For <G:

opp = HI = 12

adj = GH = 9

hyp = GI = 15

cos G = adj/hyp = 9/15 = 3/5

sin G = opp/hyp = 12/15 = 4/5

tan G = opp/adj = 12/9 = 4/3

Step-by-step explanation:

remember the triangle in a circle, when you learned about sine, cosine and the other trigonometric functions ?

this looked the same way, just that the circle was the norm-circle with radius 1.

here, now, the radius is larger, so every function line is also multiplied by the actual radius.

but the principles are the same.

sine is the up/down leg of the right-angled triangle.

cosine is the left/right leg.

tangent is sine/cosine.

I = (9, 12)

so, Pythagoras gives us the radius (line GI) :

GI² = GH² + HI² = 9² + 12² = 81 + 144 = 225

GI = sqrt(225) = 15

sin(G) × GI = HI

sin(G) × 15 = 12

sin(G) = 12/15 = 4/5 = 0.8

cos(G) × GI = GH

cos(G) × 15 = 9

cos(G) = 9/15 = 3/5 = 0.6

tan(G) = sin(G)/cos(G) = 4/5 / 3/5 = (4×5)/(5×3) =

= 4/3 = 1.333333333...

The lengths of the sides of the of the right triangles are found using the Pythagorean Theorem as follows;

Pythagorean Theorem states that the square of the length of the hypotenuse side of a right triangle is equivalent to the sum of the squares of the lengths of the legs of the right triangle.

1) The location of the angles and the congruent 90° angle and a second congruent angle indicates;

ΔABN ~ ΔTBA ~ ΔTAN

The missing values of x can be obtained using Pythagorean Theorem as follows;

2) AB = √(10² + 5²) = √(125) = 5·√5

[tex]\overline{AB}[/tex]² = (5 + x)² - [tex]\overline{AN}[/tex]²

[tex]\overline{AN}[/tex]² = (5 + x)² - [tex]\overline{AB}[/tex]²

[tex]\overline{AN}[/tex]² = 10² - x²

10² + x² = (5 + x)² - [tex]\overline{AB}[/tex]²

10² + x² = (5 + x)² - 125

[tex]\overline{AB}[/tex]² = (5 + x)² - (10² + x²) = 10·x - 75

10·x - 75 = (5·√5)² = 125

10·x - 75 = 125

10·x = 125 + 75 = 200

x = 200/10 = 20

x = 20

3) The hypotenuse side of the right triangle with sides 8 and 4 can be found as follows;

Length of the hypotenuse = √(8² + 4²) = 4·√5

Length of the leg of the larger right triangle is, length = √(8² + x²)

Therefore;

(x + 4)² = (8² + x²) + (4·√5)²

(x + 4)² - (8² + x²) = (4·√5)²

8·x - 48 = 80

8·x = 80 + 48 = 128

x = 128/8 = 16

x = 16

4) The leg of the larger right triangle = (12 + 2)² - x² = 14² - x²

14² - x² - 12² = x² - 2²

2·x² = 14² - 12² + 2² = 56

x² = 56/2 = 28

x = √(28) = 2·√7

x = 2·√7

5) The length of the shorter leg of the larger right triangle can be found as follows;

Length of the shorter leg = (20 + 25)²- x²

x² - 25² = (20 + 25)²- x² - 20²

2·x² = (20 + 25)² + 25² - 20² = 2250

x² = 2250/2 = 1125

x = √(1125) = 15·√5

x = 15·√5

6) x² - 4² = (32 + 4)² - x² - 32²

2·x² = (32 + 4)² + 4² - 32² = 288

x² = 288/2 = 144

x = √(144) = 12

x = 12

7) Let x represent the length of the right tringle and let h represent the altitude of the right triangle

(PR)² = 16² - x²

16² - x² - 12² = x² - 4²

2·x² = 16² - 12² + 4² = 128

x² = 128/2 = 64

x = √(64) = 8

The length of the short leg is; x = 8

Length of the longer leg, PR = √(16² - x²)

PR = √(16² - 8²) = 8·√3

Length of the longer leg = 8·√3

The square of the altitude = 16² - x² - 12²

Length of the altitude = √(16² - 64 - 12²) = 4·√3

8) Let x represent the length of the shorter leg, we get;

(PR)² = 18² - x²

The square of the altitude, (PS)² = 18² - x² - 15² = x² - 3²

2·x² = 18² - 15²+ 3² = 108

x² = 108/2 = 54

x = √(54) = 3·√6

Length of the shorter leg, x = 3·√6

PR = √(18² - 54) = 3·√(30)

Length of the longer leg, PR = 3·√(30)

Length of the altitude, PS = √(54 - 3²) = 3·√(15)

9) Let PQ = x, we get;

(QR)² = 30² - x²

30² - x² - (30 - 6)² = x² - 6²

2·x² = 30² - (30 - 6)² + 6² = 360

x² = 360/2 = 180

x = √(180) = 6·√5

PQ = x = 6·√5

(QR)² = 30² - 180 = 720

QR = √(720) = 12·√5

QS = √(720 - (30 - 6)²) = 12

The altitude, QS = 12

The geometric mean of 2 numbers is the square root of the product of the numbers;

10) The geometric mean of 5 and 8 = √(5 × 8) = 4·√10

11) 7 and 11

The geometric mean = √(7 × 11) = √(77)

12) 4 and 5

The geometric mean is; √(4 × 5) = 2·√5

13) 2 and 25

The geometric mean is; √(2 × 25) = √(50) = 5·√2

14) 6 and 8

The geometric mean is; √(6 × 8) = √(48) = 16·√3

15) 8 and 32

The geometric mean is; √(8 × 32) = 16

16) (15 + 5)² - y² = z²

15² + x² = y²

x² + 5² = z²

Therefore;

(15 + 5)² - 15² - x² = x² + 5²

2·x² = (15 + 5)² - 15² - 5² = 150

x² = 150/2 = 75

x = √(75) = 5·√3

x = 5·√3

z² = x² + 5²

z² = 75 + 25 = 100

z = √(100) = 10

z = 10

15² + x² = y²

15² + 75 = 300 = y²

y = √(300) = 10·√3

y = 10·√3

17) 12² - y² = z²

z² - 9² = x²

z² = 9² + x²

x² + 3² = y²

12² - x² - 3²  = z²

12² - x² - 3²  = 9² + x²

2·x² = 12² - 3² - 9² = 54

x² = 54/2 = 27

x = √(27) = 3·√3

x = 3·√3

z² = 9² + x²

z² = 9² + 27 = 108

z = 6·√3

x² + 3² = y²

y² = x² + 3²

y² = 27 + 3² = 36

y = √(36) = 6

y = 6

18) x² - 6² = y²

z² - 24² = y²

(6 + 24)² - x² = z²

Therefore; x² - 6² = z² - 24² = (6 + 24)² - x² - 24²

x² - 6² = (6 + 24)² - x² - 24² = 30² - x² - 24²

x² - 6² = 30² - x² - 24²

2·x² = 30² + 6² - 24² = 360

x² = 360/2 = 180

x = √(180) = 6·√(5)

x = 6·√(5)

z² = (6 + 24)² - x²

z² = (6 + 24)² - 180 = 720

z² = 720

z = √(720) = 12·√5

z = 12·√5

z² - 24² = y²

y² = 720 - 24² = 144

y² = 144

y = 12

19) Let x represent GH, we get;

32² - x² = (HK)²

32² - x² - (32 - 8)² = x² + 8²

2·x² = 32² - (32 - 8)² - 8² = 384

x² = 184

x = √(184) = 2·√(46)

GH = x = 2·√(46)

GH = 2·√(46)

(HK)² = (32 - 8)² + (x² - 8²)²

(HK)² = (32 - 8)² + (184 - 8²) = 696

HK = √(696) = 2·√(174)

HK = 2·√(174)

20) Let x represent the length of the lake, we get;

x² + 6² = (x + 4)² - (4² + 6²) = x² + 8·x - 36

x² + 6² = x² + 8·x - 36

8·x = 6² + 36 = 72

x = 72/8 = 9

The length of the lake, x = 9 km

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The expression when evaluated is -5

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

6(− 2/3 ) − 1.5 + ( 1/2 )

To solve the given expression, we need to apply the order of operations, which is a set of rules that dictate the order in which mathematical operations should be performed.

Using the order of operations, we can simplify the given expression as follows:

6(− 2/3 ) − 1.5 + ( 1/2 ) = -4 − 1.5 + 0.5

Evaluate

6(− 2/3 ) − 1.5 + ( 1/2 ) = -5

Hence, the solution is (b) - 5

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Answer:

C. a 55 degree angle

Step-by-step explanation:

180 degrees-125 degrees=55.

The pretax price of the item is $18. The tip is $3.60 (0.20 * 18) and the tax is $1.44 (0.08 * 18), which when added to $18 gives us a total cost of $23.04.

For a 20% tip, 8% tax sales. Total $23. 04, we can say that The pre-tax cost is $19.20.

Let x be the pre-tax cost of the sales. Then we can set up the equation:

x + 0.20x + 0.08x = 23.04

Simplifying and solving for x, we get:

1.28x = 23.04

x = 23.04 / 1.28

x = 18

The pretax price is the amount charged for a product or service before any taxes are added. It is the price that a customer would pay for a product or service without any additional charges imposed by the government. In most countries, taxes such as sales tax, value-added tax (VAT), or goods and services tax (GST) are added to the pretax price to determine the final price that a customer pays.

Pretax prices are important for businesses to determine their profit margins and pricing strategies. It is also useful for customers to compare prices between different products or services before taxes are added. However, it is important to note that the final price paid by the customer may vary based on the applicable tax rate.

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Complete Question: -

Need help on math You have a meal at a restaurant. The sale tax is 8% . You leave a tip for the waitress that is 20% of the pretax price . You spend $23.04 . What is the pretax price of the meal?

Answer:

4.5 teaspoons of vanilla

Step-by-step explanation:

We can do this by multiplying:

3/4*(6)

18/4

9/2

4.5 teaspoons of vanilla

Probability of selecting a marble at random and then put it back into the bag will be [tex]\frac{7}{36}[/tex]

The likelihood that something will occur is known as the probability.  Because we don't know how something will turn out, we might talk about the probability of one result or the potential for several outcomes. The study of events that fit into a probability distribution is known as statistics. The best example for understanding probability is flipping a coin:

There are two possible outcomes—heads or tails.

Number of purple marble=16

Number of blue marble=12

Number of white marble=14

Number of brown marble=6

The Probability is =[tex]\frac{12+14+6}{16+12+14+6}[/tex] ×  [tex]\frac{14}{16+12+14+6}[/tex]

=[tex]\frac{32}{48}[/tex] × [tex]\frac{14}{48}[/tex]

=[tex]\frac{7}{36}[/tex]

Hence, Probability of selecting a marble at random and then put it back into the bag will be      [tex]\frac{7}{36}[/tex].

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The correct answer is d. 9.9498.

To find the standard deviation of a distribution, we use the formula:

Standard deviation = √[(∑(x - mean)2)/n]

First, we need to find the mean of the distribution. The mean is the sum of all the values divided by the number of values. In this case, the mean is:

mean = (35 + 250)/5 = 57

Next, we need to find the sum of the squared differences between each value and the mean. This is:

∑(x - mean)2 = (35 - 57)2 + (250 - 57)2 = 485 + 37249 = 37734

Finally, we divide this sum by the number of values and take the square root to get the standard deviation:

Standard deviation = √(37734/5) = √7546.8 = 9.9498

So the standard deviation of the distribution is 9.9498.

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Answer: D

Step-by-step explanation:

7x + 4 = 88

x = 12

angle 1 = 180 - 7(12) + 2

angle 1 = 98

So it's D

Answer: The angle measure for angle 1 would be 98 degrees.

Step-by-step explanation: First, you set 88 equal to (7x+4) since they are vertical angles. (Vertical angles are congruent.) Then you solve and find that x=12. You then plug 12 in to the second equation, (7x-2) and get 82. Since angle 1 is supplementary to the angle that measures 82 degrees, you would take 180-82= Angle 1.

Answer:

0.03125

Step-by-step explanation:

[tex]a_{n}[/tex] = [tex]a_{1}[/tex][tex](r)^{n-1}[/tex]  

[tex]a_{7}[/tex] = (2)[tex]\frac{1}{2} ^{(7-1)}[/tex]

[tex]a_{7}[/tex] = (2)[tex]\frac{1}{2} ^{6}[/tex]

[tex]a_{7}[/tex] = 0.03125

Helping in the name of Jesus.

Answer:

To show that T<0,2> (0,1), we need to evaluate the transformation T at the point (0,1) and check if the result is in the range of T.

T<0,2> (0,1) means that the transformation T takes the point (0,1) in the input space to a point in the output space that has coordinates (0,2).

Let's evaluate T at (0,1):

T<0,2> (0,1) = D3(0,1) + (0,1)

= (0+0, 3+1)

= (0, 4)

The output point (0,4) is not equal to (0,2), which means that T<0,2> (0,1) is false. Therefore, T<0,2> does not map the point (0,1) to (0,2).

To determine if T<0,2> (x,y) = D3 (x,y) is a true statement for any point (x,y), we need to check if the transformation T always equals the function D3.

T<0,2> (x,y) = (x, 3+y)

D3(x,y) = (x,y,3)

Since T and D3 have different ranges (T has a range of R^2 and D3 has a range of R^3), the statement T<0,2> (x,y) = D3(x,y) is not true for any point (x,y).

Therefore, we cannot equate T<0,2> and D3, and the statement T<0,2> (x,y) = D3 (x,y) is false for any point (x,y).

Answer:

15%

Step-by-step explanation:

To find the scale factors with fractions, you first need to understand what scale factors are. Scale factors are the ratios of corresponding lengths in two similar figures. They tell you how much larger or smaller one figure is compared to another.

To simplify fractions, you need to find the greatest common factor (GCF) of the numerator and denominator and divide them by it.

For example, if we have the fraction 6/12, we can simplify it by finding the GCF of 6 and 12, which is 6. Then, we divide both the numerator and denominator by 6:

6/12 = (6 ÷ 6) / (12 ÷ 6) = 1/2

So, 6/12 simplifies to 1/2.

If the fraction is already in its simplest form, then there is no need to simplify it further.

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The complete question i:

How do you find the scale factors with fractions? Give example.

Answer:

x = 10

Step-by-step explanation:

Since KL is parallel to HI

m∠JKL = m∠JHI

m∠JLK = m∠LIH

For the two triangles JKL and JHI, we have the common angle J and two angles equal to the corresponding two angles

Therefore the triangles are similar

Ratio of the sides to the corresponding sides must be equal

Hence
[tex]\dfrac{JH}{JK} = \dfrac{JI}{JL}\\[/tex]

JH = JK + KH = 28 + 20 = 48

JI = JL + JL = 14 + x

Therefore
[tex]\dfrac{JH}{JK} = \dfrac{JI}{JL}\\\\= > \dfrac{48}{28} = \dfrac{14 + x}{14}[/tex]

[tex]\dfrac{48}{28} = \dfrac{12}{7}[/tex]  by dividing numerator and denominator by 4

Hence

[tex]\dfrac{48}{28} = \dfrac{14 + x}{14}\\\\\rightarrow \quad \dfrac{12}{7} = \dfrac{14 + x}{14}\\\\[/tex]

Multiply both sides by 14:
[tex]14 \cdot \dfrac{12}{7} = 14 \cdot \dfrac{14 + x}{14}\\\\2 \cdot 12 = 14 + x\\\\24 = 14 + x\\\\\text{Subtract 14 both sides:}\\\\24 - 14 = x\\\\or\\\\x = 10\\\\[/tex]

Step-by-step explanation:

The area A of a rectangle is given by multiplying its length and width. In this case, the length is 7x - 1 units and the width is 2x + 3 units. Therefore, the area of the rectangle is:

A = (7x - 1)(2x + 3)

= 14x^2 + 19x - 3

Hence, the area of the rectangle is 14x^2 + 19x - 3 square units.

Answer:

14x^2+19x-3

Step-by-step explanation:

you have to times them together so

(7x-1)(2x+3)

14x^2+21x-2x-3

=14x^2+19x-3

To model the path of the rock, we can use a quadratic equation of the form y = ax^2 + bx + c, where y is the height of the rock, x is the distance from the catapult, and a, b, and c are constants.
We can use the given information to find the values of a, b, and c.

First, we know that the rock launches out of the catapult at a height of 0 feet, so when x = 0, y = 0. This means that c = 0.
Next, we know that the rock reaches its maximum height of 50 feet when it is 75 feet away from the catapult, so when x = 75, y = 50. We can plug these values into the equation and simplify:
50 = a(75)^2 + b(75) + 0
50 = 5625a + 75b

Finally, we know that the rock hits the wall 8 feet from the top when it is 120 feet away from the catapult, so when
x = 120, y = 40 - 8 = 32. We can plug these values into the equation and simplify:
32 = a(120)^2 + b(120) + 0
32 = 14400a + 120b
We now have a system of two equations with two unknowns:
5625a + 75b = 50
14400a + 120b = 32

We can use substitution or elimination to solve for a and b. Using elimination, we can multiply the first equation by -1.6 to eliminate the b term:
-9000a - 120b = -80
14400a + 120b = 32
Adding the two equations together gives:
5400a = -48
Solving for a gives:
a = -48/5400 = -0.00888888889

We can then plug this value of a back into one of the original equations to solve for b:
5625(-0.00888888889) + 75b = 50
-50 + 75b = 50
75b = 100
b = 100/75 = 1.33333333333

So the equation that models the path of the rock is:
y = -0.00888888889x^2 + 1.33333333333x + 0
Or, rounding to three decimal places:
y = -0.009x^2 + 1.333x

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A. If a friend's kid has no curfew, what is the probability that they don't have chores? B. the probability that a friend's kid doesn't have chores given that they have no curfew is 1/3. C. I chose this question because it involves calculating a conditional probability based on a given set of data.

Probability can also be used to describe more complex events, such as the likelihood of a stock price increasing by a certain amount in a given period of time, or the probability of a medical treatment being effective for a certain disease.

My Conditional Frequency Question is:

If a friend's kid has no curfew, what is the probability that they don't have chores?

Solution:

The conditional probability of a kid not having chores given that they have no curfew can be found using the formula:

P(No Chores | No Curfew) = P(No Chores and No Curfew) / P(No Curfew)

From the table, we can see that the number of kids who have no curfew and no chores is 2, and the total number of kids who have no curfew is 6. Therefore:

P(No Chores | No Curfew) = 2/6 = 1/3

So the probability that a friend's kid doesn't have chores given that they have no curfew is 1/3.

I chose this question because it involves calculating a conditional probability based on a given set of data. This is a common type of question in statistics and probability, and it requires an understanding of the basic concepts of probability such as conditional probability and independence. The table provided gives us the data we need to calculate the conditional probability, and the solution involves applying the formula for conditional probability and simplifying the fraction.

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sin(θ + ɸ) = 17/145 and cos(θ + ɸ) = 144/145.

Suppose sin θ = -3/5, sin ɸ = 20/29. Moreover, suppose θ is in Quadrant IV and ɸ is in Quadrant l. We can find sin(θ + ɸ) and cos(θ + ɸ) by using the following formulas: sin(θ + ɸ) = sin θ cos ɸ + cos θ sin ɸ and cos(θ + ɸ) = cos θ cos ɸ - sin θ sin ɸ.

First, we need to find cos θ and cos ɸ. Since θ is in Quadrant IV, cos θ is positive. We can use the Pythagorean identity, sin² θ + cos² θ = 1, to find cos θ:

cos² θ = 1 - sin² θ
cos² θ = 1 - (-3/5)²
cos² θ = 1 - 9/25
cos² θ = 16/25
cos θ = √(16/25)
cos θ = 4/5

Similarly, since ɸ is in Quadrant l, cos ɸ is also positive. We can use the Pythagorean identity to find cos ɸ:

cos² ɸ = 1 - sin² ɸ
cos² ɸ = 1 - (20/29)²
cos² ɸ = 1 - 400/841
cos² ɸ = 441/841
cos ɸ = √(441/841)
cos ɸ = 21/29

Now we can plug in the values of sin θ, sin ɸ, cos θ, and cos ɸ into the formulas for sin(θ + ɸ) and cos(θ + ɸ):

sin(θ + ɸ) = sin θ cos ɸ + cos θ sin ɸ
sin(θ + ɸ) = (-3/5)(21/29) + (4/5)(20/29)
sin(θ + ɸ) = -63/145 + 80/145
sin(θ + ɸ) = 17/145

cos(θ + ɸ) = cos θ cos ɸ - sin θ sin ɸ
cos(θ + ɸ) = (4/5)(21/29) - (-3/5)(20/29)
cos(θ + ɸ) = 84/145 + 60/145
cos(θ + ɸ) = 144/145

Therefore, sin(θ + ɸ) = 17/145 and cos(θ + ɸ) = 144/145.

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