High school students from grades 9–10 and 11–12 were asked to choose the kind of band to have play at a school dance: rap, rock, or country. Their choices were as follows: Grades 9–10—Rap: 40; Rock: 30; Country: 55 Grades 11–12—Rap: 60; Rock: 25; Country: 35 Which of the following is a correct two-way frequency table for the data? Band preference for school dance Rap Rock Country Row totals Grades 9–10 0. 28 0. 21 0. 38 0. 86 Grades 11–12 0. 41 0. 17 0. 24 0. 83 Column totals 0. 69 0. 38 0. 62 1 Band preference for school dance Rap Rock Country Row totals Grades 9–10 0. 40 0. 55 0. 61 0. 86 Grades 11–12 0. 60 0. 45 0. 39 0. 83 Column totals 1 1 1 1 Band preference for school dance Rap Rock Country Row totals Grades 9–10 40 30 55 125 Grades 11–12 60 25 35 120 Column totals 100 55 90 245 Band Preference for School Dance Rap Rock Country Row totals Grades 9–10 0. 32 0. 24 0. 44 1 Grades 11–12 0. 50 0. 21 0. 29 1 Column totals 0. 69 0. 38 0. 62 1.

On solving the provided question, we can say that from the provided graphs, we have Domain = [(-4, 0) to  (-3, 1)]

Mathematicians  use the graphs to logically convey facts or values using the visual representations or charts. A graph point will typically reflect a relationship between two or more things. Nodes, or vertices, and edges make form a graph, a non-linear data structure. Glue together the nodes, often referred to as vertices. This graph has vertices V=1, 2, 3, 5, and edges E=1, 2, 1, 3, 2, 4, and (2.5), (3.5). (4.5). Statistical charts (bar charts, pie charts, line charts, etc.) graphical representations of exponential growth. a logarithmic graph in the shape of a triangle

from the provided graphs, we have  

domain of the function

Domain = [(-4, 0) to  (-3, 1)]

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Answer:

-648

Step-by-step explanation:

Answer:

-648

Step-by-step explanation:

(10 times 10) equals 100

(-34 times 22) equals -748

then add them both together

Answer:

A

Step-by-step explanation:

you do 3.6 times both the cm or number and you get 14.4 cm and 5.4 cm

The solution of covariance and correlation between random variables X and Y given by

Cov(X, Y) = 75

Corr(X, Y) = \(\frac{75}{125}\) = 0.6

To find covariance and correlation between random variables X and Y,

we'll use the following formulas:

Cov(X, Y) = E(XY) - E(X)E(Y)

Corr(X, Y) = Cov(X, Y) / \(\sqrt{((Var(X)}\)) × \(\sqrt{(Var(Y)}\)))

Given:

E(X) = 4

E(\(X^2\)) = 41

E(Y|X) = 1 + 3X

Var(Y) = 625

Let's calculate each step:

Step 1: Find E(XY)

E(XY) = E(E(XY|X))  [Law of Total Expectation]

E(XY) = E(X(E(Y|X)))  [Substituting Y = E(Y|X)]

E(XY) = E(X(1 + 3X))  [Substituting E(Y|X) = 1 + 3X]

E(XY) = E(X + \(3X^2\))  [Expanding]

E(XY) = E(X) + 3E\((X^2)\)  [Linearity of Expectation]

E(XY) = 4 + 3 × 41  [Substituting E(X) = 4 and E(\(X^2\)) = 41]

E(XY) = 4 + 123

E(XY) = 127

Step 2: Find Cov(X, Y)

Cov(X, Y) = E(XY) - E(X)E(Y)

Cov(X, Y) = 127 - 4 × E(Y)  [Substituting E(X) = 4]

Cov(X, Y) = 127 - 4 × E(E(Y|X))  [Law of Total Expectation]

Cov(X, Y) = 127 - 4 × E(1 + 3X)  [Substituting Y = E(Y|X)]

Cov(X, Y) = 127 - 4 × (1 + 3 × E(X))  [Linearity of Expectation]

Cov(X, Y) = 127 - 4 × (1 + 3 × 4)  [Substituting E(X) = 4]

Cov(X, Y) = 127 - 4 × (1 + 12)

Cov(X, Y) = 127 - 4 × 13

Cov(X, Y) = 127 - 52

Cov(X, Y) = 75

Step 3: Find Corr(X, Y)

Corr(X, Y) = Cov(X, Y) / (\(\sqrt{(Var(X)}\)) × \(\sqrt{(Var(Y)}\)))

Corr(X, Y) = 75 / ( \(\sqrt{(Var(X\))  × \(\sqrt{(Var(Y)}\)))

[Substituting Cov(X, Y) = 75]

Corr(X, Y) = 75 / (\(\sqrt{(E(X^{2} ) - E(X)^{2} }\)× \(\sqrt{(Var(Y)}\)))

 [Substituting Var(X) = E(\(X^2\)) - E\(X^2\)]

Corr(X, Y) = 75 / (\(\sqrt{41-16}\) × \(\sqrt{625}\))

 [Substituting E(X) = 4, E(\(X^2\)) = 41, Var(Y) = 625]

Corr(X, Y) = 75 / (\(\sqrt{41-16}\) × 25)

Corr(X, Y) = 75 / (\(\sqrt25}\)× 25)

Corr(X, Y) = 75 / (5 × 25)

Corr(X, Y) = 75/125 = 0.6

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6 miles , is the one additional step which Cecil must travel in order to return to starting point A . His move accord expression is

8+(-5)+7+6+(-7)+5+(-8)+6

Let Cecil start from a point A and move toward point B . His movement expression is

8+(-5)+7+6+(-7)+5+(-8)

A →8 <--5 →7→6<--7→5<--8 , here  shows forward steps and shows backward steps

A→8→7→6→5 (B)

Total distance travelled by Cecil from A to B = 26 miles

B→5→7→8

Total distance travelled by Cecil from B to A = 20 miles

Cecil wants to return to the starting point A ,so he must be travel same distance as travelled from A to B i.e. 26 miles .

But Cecil travel from B to A is 20 and not 26 . The difference between the distance travelled by Cecil is (A→B – B→A) = 26-20 = 6 miles

Hence, Cecil required travel distance is 6 miles to return his starting point A.

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The percent of students in grade 8 those who like to join dance club is equal to 25%.

Total number of students in grade 8 = 20

Total number of students of grade 8  join in dance club = 5

Percent of students of grade 8 prefer to join dance club

= ( total number of students prefer dance club ) / ( Total number of students in grade 8 ) × 100

Substitute the value in the formula we get,

⇒ Percent of students of grade 8 prefer to join dance club

= ( 5 ) / ( 20 ) × 100

⇒ Percent of students of grade 8 prefer to join dance club  = 0.25 × 100

⇒ Percent of students of grade 8 prefer to join dance club  = 25%

Therefore, percent of grade 8 students who preferred to join dance club is equal to 25%.  

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The above question is incomplete, the complete question is:

What percent of students in grade 8 prefer to join a dance club?

                      Dance club      Hiking club     Total

Grade 7               15                       15                 30

Grade 8               5                        15                  20

Total                    20                     30                    50

Answer:

$111.5699. Round to hundredths $111.57

Step-by-step explanation:

First identify the %

20% =.20 20/100= .20

3.5% = .035 3.5/100= .035

Then Find the tax rate and tip

Tax Rate .035 × 90.34

= $3.1619 (tax rate)

Tip. is .20× 90.34

= $18.068 (tip)

Final Add all it up tax rate+ tip + original price

$90.34. Don't forget to line up your demicals

$3.1619. or use a calculator

+ $18.068

Total = $111.5699

round to the nearest hundredth

$111.57 is your total

Answer:

1: C) y = -x + 1

2: D) y = -3x + 8

Step-by-step explanation:

Well number 1,

c is the correct option because,

-2 -> 2

2+1 - 3

This rule applies for all the x values.

2,

X: -2, -1, 0, 1, 2

Y: 14, 11, 8, 5, 2

d is the correct option because -3*-2 is 6 + 8 = 14.

this rule applies for all x values.

Thus,

the answers are C and D.

Hope this helps :)

The solution to the equation 3x(X + 6) = -10 using the most direct method is:
First, distribute the 3x: 3x^2 + 18x = -10
Next, move all terms to one side: 3x^2 + 18x + 10 = 0

Now, use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a
Where a = 3, b = 18, and c = 10.
Plugging in these values, we get:x = (-18 ± √(18^2 - 4(3)(10))) / 2(3)
x = (-18 ± √264) / 6
x = (-18 ± 2√66) / 6
x = (-3 ± √66) / 1
Therefore, the solution to the equation is: x = -3 + √66 or x = -3 - √66

Most simplified form and there are two solutions, written using the + symbol. To solve the equation 3x(X + 6) = -10, first distribute the 3x across the parentheses.
Answer:
1. 3x^2 + 18x = -10
Next, move all terms to one side to set the equation equal to zero.
2. 3x^2 + 18x + 10 = 0
Lastly, solve the quadratic equation for x. In this case, we can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. Here, a = 3, b = 18, and c = 10.
3. x = (-18 ± √(18^2 - 4(3)(10))) / (2 * 3)
Simplify the expression and calculate the two solutions for x.

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To solve the equation 3x(x + 6) = -10, we first distribute the 3x to get 3x^2 + 18x = -10, then add 10 to both sides to set the equation to equal 0, and finally use the quadratic formula to find the values of x.

The equation given to us is: 3x(x + 6) = -10. The first step is to distribute the 3x across the x and 6 to get 3x^2 + 18x = -10. The next step is to add 10 to both sides to make it equal to zero. This gives us 3x^2 + 18x + 10 = 0. From here, we use the quadratic formula, x = [-b ± sqrt(b^2 - 4ac)] / (2a) where a = 3, b = 18, and c = 10. The solution to the equation are the values of x that satisfy this equation.

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Answer:

-3f+8

Step-by-step explanation:

you have to look at the different sings of the numbers. you have (+) × (-) + (+)

The exact value  of \(sin 2x\) is \(4√5/9.\)

Given that \(sec x = 3/2 and csc y = 3\)where x and y are in the 2x = 2 sin x quadrant, we need to find the exact value of sin 2x.

In the first quadrant, we have the following values of the trigonometric ratios:\(cos x = 2/3 and sin y = 3/5\)

Also, we know that sin \(2x = 2 sin x cos x.\)

Now, we need to find sin x.

Having sec x = 3/2, we can use the Pythagorean identity

\(^2x + 1 = sec^2xtan^2x + 1 = (3/2)^2tan^2x + 1 = 9/4tan^2x = 9/4 - 1 = 5/4tan x = ± √(5/4) = ± √5/2\)

As x is in the first quadrant, it lies between 0° and 90°.

Therefore, x cannot be negative.

Hence ,\(tan x = √5/2sin x = tan x cos x = √5/2 * 2/3 = √5/3\)

Now, we can find sin 2x by using the value of sin x and cos x derived above sin \(2x = 2 sin x cos xsin 2x = 2 (√5/3) (2/3)sin 2x = 4√5/9\)

Therefore, the exact value  of sin 2x is 4√5/9.

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Answer:

\(\frac{1}{6} a-\frac{1}{6}\)

Step-by-step explanation:

\(-\frac{2}{3} a+\frac{5}{6}a-\frac{1}{6}\)

\(=-\frac{4}{6} a+\frac{5}{6}a -\frac{1}{6}\)

\(= \frac{1}{6} a-\frac{1}{6}\)

Answer:

\( \frac{( - 2)}{3} a + \frac{5}{6}a - \frac{1}{6} \\ = \frac{(- 4a) + 5a - 1}{6} = \frac{(a - 1)}{6} \)

hope this helps.

Answer:

y=9

Step-by-step explanation:

5(y-10)=-5

We move all terms to the left:

5(y-10)-(-5)=0

We add all the numbers together, and all the variables

5(y-10)+5=0

We multiply parentheses

5y-50+5=0

We add all the numbers together, and all the variables

5y-45=0

We move all terms containing y to the left, all other terms to the right

5y=45

y=45/5

y=9

The value of y = 9 is correct. Therefore, the solution is y = 9.

To solve the equation 5(y - 10) = -5 for y, follow these steps:

Step 1: Distribute the 5 on the left side of the equation:

5y - 50 = -5

Step 2: Move the constant term to one side by adding 50 to both sides:

5y - 50 + 50 = -5 + 50

Step 3: Simplify the left side and evaluate the right side:

5y = 45

Step 4: Isolate y by dividing both sides by 5:

5y / 5 = 45 / 5

Step 5: Simplify and calculate y:

y = 9

The solution to the equation is y = 9. To verify, substitute y with 9 in the original equation:

5(9 - 10) = -5

5(-1) = -5

-5 = -5

Since both sides are equal, the value of y = 9 is correct. Therefore, the solution is y = 9.

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The slope (m) of the line is 19 and the y-intercept (b) is -4. The equation of the line can be expressed as y = 19x - 4.

The slope (m) of a straight line can be calculated using the formula:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are the coordinates of two points on the line.

Using the given points (3, 53) and (5, 91), we can substitute the values into the formula:

m = (91 - 53) / (5 - 3)

m = 38 / 2

m = 19

Therefore, the slope (m) of the straight line is 19.

To determine the y-intercept (b), we can use the slope-intercept form of a linear equation:

y = mx + b

where m is the slope and b is the y-intercept.

Using the point (3, 53) and the slope we just calculated (m = 19), we can substitute the values into the equation:

53 = 19(3) + b

53 = 57 + b

Now, solving for b:

b = 53 - 57

b = -4

Therefore, the y-intercept (b) of the straight line is -4.

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Answer:

I'm not sure sorry if I'm wrong but I think the answers would be 3,4,5

Solution:

The distance between two points can be calculated by the formula

Answer:

170

Step-by-step explanation:

To show that | | | | | 1 1 1 a b c a 2 b 2 c 2 | | | | | = (b − a)(c − a)(c − b), we can use row reduction. We start by subtracting the first row from the second row and the first row from the third row, which gives:

| | | | | 1 1 1 a b c a 2 b 2 c 2 | | | | |

R2 - R1 | | | | | 1 0 b-a 0 b-a c-a a 2 b 2 c 2 | | | | |

R3 - R1 | | | | | 1 0 b-a 0 b-a c-a 0 b 2(c-a) | | | | |

Next, we multiply the second row by (c-a) and the third row by b-a, which gives:

| | | | | 1 0 b-a 0 b-a c-a a 2 b 2 c 2 | | | | |

(c-a)R2 | | | | | c-a 0 (b-a)(c-a) 0 (b-a)(c-a)(c-b) (c-a)a (c-a)2b (c-a)2c 2(c-a)2bc | | | | |

(b-a)R3 | | | | | b-a 0 (b-a)(c-a) 0 (b-a)(c-a)(c-b) 0 b(b-a) 2bc(b-a) (c-a)b2 | | | | |

Finally, we add (b-a) times the second row to the third row, which gives:

| | | | | c-a 0 (b-a)(c-a) 0 (b-a)(c-a)(c-b) (c-a)a (c-a)2b (c-a)2c 2(c-a)2bc | | | | |

(b-a)R3 | | | | | 0 0 (b-a)(c-a) 0 (b-a)(c-a)(c-b) b(b-a) 2bc(c-a) (c-a)b2+(b-a)2c | | | | |

Now, we can see that the determinant of the matrix is the same as the determinant of the last row, which is:

(b-a)(c-a)(c-b)(c-a)b2+(b-a)2c = (b-a)(c-a)(c-b)c

Therefore, we have shown that | | | | | 1 1 1 a b c a 2 b 2 c 2 | | | | | = (b − a)(c − a)(c − b).

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a) The relevant population is the heavy soft-drink users in the given case, and the appropriate sampling design that should be used is  stratified random sampling. The list of all heavy soft-drink users is the sampling frame.

b) Cluster sampling refers to a sampling design where population is divided into naturally occurring groups and a random sample of clusters is chosen.

The advantages are efficient, easy to perform, and used when the population is widely dispersed. The disadvantages are sampling errors, have lower level of precision, and have the standard error of the estimate.

a) The relevant population for the public relations research department to investigate the initial reactions of heavy soft-drink users to a new all-natural soft drink is heavy soft-drink users. The appropriate sampling design that can be used to investigate the issues is stratified random sampling.

Stratified random sampling is a technique of sampling in which the entire population is divided into subgroups (or strata) based on a particular characteristic that the population shares. Then, simple random sampling is done from each stratum. Stratified random sampling is appropriate because it ensures that every member of the population has an equal chance of being selected.

Moreover, it ensures that every subgroup of the population is adequately represented, and reliable estimates can be made concerning the entire population. The list of all heavy soft-drink users can be the sampling frame.

b) Cluster sampling is a type of sampling design in which the population is divided into naturally occurring groups or clusters, and a random sample of clusters is chosen. The elements within each chosen cluster are then sampled.

The advantages of cluster sampling are:

The disadvantages of cluster sampling are:

A situation where cluster sampling would be appropriate is in investigating the effects of a new medication on various groups of people. In this case, the population can be divided into different clinics, and a random sample of clinics can be selected. Then, all patients who meet the inclusion criteria from the selected clinics can be recruited for the study. This way, the study will be less expensive, and it will ensure that the sample is representative of the entire population.

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The point of intersection of two straight line graphs is the point with coordinates that satisfies both lines equations

The length of segment EF is presented as follows;

\(EF =\underline{\mathbf{ 17\dfrac{1}{7} \ ft.}}\)

The reason why the above value is correct is as follows;

The given parameters are;

Height of right triangle ACD = 40 ft.

Height of right triangle BCD = 30 ft.

Required:

To find the length of EF

Solution:

The equation of the lines AC and BD are found and equated to find the height of EF as follows;

The slope of AC = \(\dfrac{-40}{CD}\)

The equation of AC  is presented as follows;

\(y - 40 = \dfrac{-40}{CD} \times (x - 0)\)

\(y = \dfrac{-40}{CD} \times x + 40\)

The slope of BD = \(\dfrac{30}{CD}\)

The equation of BD is given as follows;

\(y - 30 = \dfrac{30}{CD} \times (x - CD)\)

\(y = \dfrac{30}{CD} \times (x - CD)+ 30\)

Equating both values of y to find the value of y at the intersecting point E, gives;

\(\dfrac{-40}{CD} \times x + 40 = \dfrac{30}{CD} \times (x - CD)+ 30\)

Which gives;

\(\dfrac{40 \cdot CD - 40 \cdot x}{CD} = \dfrac{30 \cdot x}{CD}\)

Therefore;

40·CD - 40·x = 30·x

\(CD = \dfrac{70\cdot x}{40} = \dfrac{7\cdot x}{4}\)

\(CD = \dfrac{7\cdot x}{4}\)

Therefore, at E, we have;

\(y = EF= \dfrac{40 \cdot CD - 40 \cdot x}{CD} = \dfrac{30 \cdot x}{CD}\)

\(y = EF = \dfrac{40 \times \dfrac{7}{4}\cdot x - 40 \cdot x}{\dfrac{7}{4}\cdot x} = \dfrac{30 \cdot x}{\dfrac{7}{4}\cdot x} = \dfrac{120}{7} = 17\dfrac{1}{7}\)

\(\underline {EF = 17\dfrac{1}{7} \ ft.}\)

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The 4 tests for similar triangles are:-

AAA: Three pairs of equal angles.

SSS: Three pairs of sides in the same ratio.

SAS: Two pairs of sides in the same ratio and an equal included angle.

ASA: Two angles and the side included between the angles of one triangle are equal

What is AAA,SAS,ASA,SSS?

According to the SSS rule, two triangles are said to be congruent if all three sides of one triangle are equal to the corresponding three sides of the second triangle. 

According to the SAS rule, two triangles are said to be congruent if any two sides and any angle between the sides of one triangle are equal to the corresponding two sides and angle between the sides of the second triangle.

According to the ASA rule, two triangles are said to be congruent if any two angles and the side included between the angles of one triangle are equal to the corresponding two angles and side included between the angles of the second triangle.

According to the  AAA rule, "if in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion), and hence the two triangles are identical."

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Answer:

The answer is

\(y = - \frac{5}{2} x - 1\)

Step-by-step explanation:

If two equations are parallel it means that both their gradients are equal. We were given the equation:

\(5x + 2y = 12\)

in order to find the gradient of this equation that we are given we have to ensure that it is in its simplest form:

\(5x + 2y = 12 \\ 2y = 12 - 5x \\ \frac{2y}{2} = \frac{12}{2} - \frac{5}{2} x \\ y = 6 - \frac{5}{2} x \: \\ or \: \\ y = - \frac{5}{2} x + 6\)

Therefore the gradient of the parallel line with points (-2, 4) is also -5/2

\(y = mx + c \\ 4 = - \frac{ 5}{2} ( - 2) + c \\ 4 = \frac{10}{2} + c \\ 4 = 5 + c \\ 4 - 5 = c \\ \)

\(c = - 1 \\ hence \: the \: equation \: for \: the \: \\ parallel \: line \: is \\ \: y = - \frac{5}{2} x - 1\)

Answer:

it should be A, but it depends on how you look at it

Answer:

3. w=P/2-L

Step-by-step explanation:

Given formula:

Solving for w:

Correct answer choice is:

Here are the vectors **t**, **n**, and **b** at the given point:

* **t** = (-3 sin t, 3 cos t, 0)

* **n** = (-3 cos t, -3 sin t, 3 / cos^2 t)

* **b** = (3 cos^2 t, -3 sin^2 t, -3)

The vector **t** is the unit tangent vector, which points in the direction of the curve at the given point. The vector **n** is the unit normal vector, which points in the direction perpendicular to the curve at the given point. The vector **b** is the binormal vector, which points in the direction that is perpendicular to both **t** and **n**.

To find the vectors **t**, **n**, and **b**, we can use the following formulas:

```

t(t) = r'(t) / |r'(t)|

n(t) = (t(t) x r(t)) / |t(t) x r(t)|

b(t) = t(t) x n(t)

```

In this case, we have:

```

r(t) = (3 cos t, 3 sin t, 3 ln cos t)

r'(t) = (-3 sin t, 3 cos t, 3 / cos^2 t)

```

Substituting these into the formulas above, we can find the vectors **t**, **n**, and **b** as shown.

The vectors **t**, **n**, and **b** are all orthogonal to each other at the given point. This is because the curve is a smooth curve, and the vectors are defined in such a way that they are always orthogonal to each other.

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The binormal vector (b) is perpendicular to both the tangent and normal vectors and completes the orthogonal coordinate system.

To find the vectors t, n, and b at the given point, we need to calculate the first derivative, second derivative, and third derivative of the position vector r(t).
Given r(t) = (3 cos t, 3 sin t, 3 ln cos t), we can calculate the derivatives as follows:
First derivative:
r'(t) = (-3 sin t, 3 cos t, -3 sin t / cos t)
Second derivative:
r''(t) = (-3 cos t, -3 sin t, -3 cos t / cos^2 t + 3 sin^2 t / cos t)
       = (-3 cos t, -3 sin t, -3 cos t / cos^2 t + 3 tan^2 t)
Third derivative:
r'''(t) = (3 sin t, -3 cos t, 6 cos t / cos^3 t - 6 sin t / cos t)
        = (3 sin t, -3 cos t, 6 sec^3 t - 6 tan t sec t)
At the given point (3, 0, 0), substitute t = 0 into the derivatives to find the vectors:
r'(0) = (0, 3, 0)
r''(0) = (-3, 0, 3)
r'''(0) = (0, -3, 6)
Therefore, at the given point, the vectors t, n, and b are:
t = r'(0) = (0, 3, 0)
n = r''(0) = (-3, 0, 3)
b = r'''(0) = (0, -3, 6)
These vectors represent the tangent, normal, and binormal vectors, respectively, at the given point.
The tangent vector (t) represents the direction of motion of the curve at that point. The normal vector (n) is perpendicular to the tangent vector and points towards the center of curvature.

The binormal vector (b) is perpendicular to both the tangent and normal vectors and completes the orthogonal coordinate system.
Remember to check your calculations and units when applying this method to different functions.

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Answer:

B

Step-by-step explanation:

(g - f)(x)

= g(x) - f(x)

= log(x - 3) + 6 - (\(\sqrt[3]{12x+1}\) + 4)

= log(x - 3) + 6 - \(\sqrt[3]{12x+1}\) - 4 ← collect like terms

= log(x - 3) - \(\sqrt[3]{12x+1}\) + 2

Value of the given function\((g-f)x = log(x-3)-\sqrt[3]{12x+1} +2\).

" Function is defined as the relation between the given variables is such that every input has exactly one output each."

According to the question,

Given functions are,

\(f(x) = \sqrt[3]{12x+1} +4\\\\g(x) = log(x-3) +6\)

Substitute the value in the function (g -f)x we get,

\((g-f)x = g(x) -f(x)\)

             \(= log(x-3) + 6 -( \sqrt[3]{12x+1} +4)\\\\= log(x-3) + 6 -\sqrt[3]{12x+1} -4\\\\= log(x-3) -\sqrt[3]{12x+1} +2\)

Hence, Option(B) is the correct answer.

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Based on the given clues, we can determine the person, drink, and price paid for each individual:

Calvin: Root Beer, $4.99

Dean: Lemonade, $7.99

Kelli: Water, $6.99

Lee: Iced Tea, $5.99

From clue 1, we know that either Calvin or the person who got the Root Beer paid $4.99. Since Calvin paid more than Lee according to clue 4, Calvin cannot be the one who got the Root Beer. Therefore, Calvin paid $4.99.

From clue 2, Kelli paid $6.99.

From clue 3, the person who got the water paid $1 less than Dean. Since Dean paid the highest price, the person who got the water paid $1 less, which means Lee paid $5.99.

From clue 5, the person who got the Root Beer paid $1 less than the person who got the Iced Tea. Since Calvin got the Root Beer, Lee must have gotten the Iced Tea.

Therefore, the final assignments are:

Calvin: Root Beer, $4.99

Dean: Lemonade, $7.99

Kelli: Water, $6.99

Lee: Iced Tea, $5.99

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To solve for the diameter and radius of a cone with a volume of 8 and a height of 6, we need to use the formulas for the volume and surface area of a cone.

The volume of a cone is given by the formula:

V = 1/3 * π * r^2 * h

where V is the volume, r is the radius, h is the height, and π is the mathematical constant pi (approximately 3.14).

We know that the volume is 8 and the height is 6, so we can plug these values into the formula and solve for the radius:

8 = 1/3 * π * r^2 * 6

r^2 = 8/(π*6/3)

r^2 = 4/π

r = √(4/π)

r ≈ 0.798

The radius is approximately 0.798.

To find the diameter, we simply multiply the radius by 2:

d = 2 * r

d ≈ 1.596

Therefore, the diameter is approximately 1.596 and the radius is approximately 0.798.