Two points on the path of a planet are A and B. The points A and B have coordinates (1, 4, 2) and (2, –1, 3) respectively.
The line l has equation r= (2i-j+3k)+µ(i-j+k)
(a) Calculate the distance between the points A and B .
(b) The line AB makes an acute angle θ with l. Calculate θ.
(c) The point P on the line l is where λ = p.
(i) for the vector AP how that :
AP. (i-j+k)=7+3p
(ii) Hence find the coordinates of the foot of the perpendicular from the point
A to the line l.
(c) Determine the cartesian equation of line l.
(d) Determine the cartesian equation of line A

Answer:

the function f(x) is f(x) = 5x + 4.

Step-by-step explanation:

To find f(x), we need to use the formula:

(gof)(x) = g(f(x)) = 3f(x) - 2

We are given that (gof)(x) = 15x + 10, so we can substitute this expression into the formula to get:

3f(x) - 2 = 15x + 10

Simplifying this equation, we get:

3f(x) = 15x + 12

Dividing both sides by 3, we get:

f(x) = 5x + 4

Therefore, the function f(x) is f(x) = 5x + 4.

Answer:

=x2-100

Step-by-step explanation:

When I said x2 I mean (x*x)

The age of the 7th volunteer has to be 602 years so that the range of the all volunteer ages will be 557.

Given ages of 6 volunteers = (56, 81, 54, 47, 45, 94)

The range = maximum value - minimum value.

So, range of the given list = 94 - 45 = 49.

To increase, the range to 557, we have to add an age that is greater than 94, To obtain that age we have to solve 557 - 45 = 508. [here 557 is the maximum value]

Now, by adding the obtained range to the 94 we can get the required age of 7th volunteer. So 508 + 94 = 602.

From the above explanation, we can conclude that the age of the 7th volunteer has to be 602 years which is unrealistic so that the range becomes 557.

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Answer:

1, -9

Step-by-step explanation:

xy = -9

x + y = -8    solve this for x and substitute into the 1st equation

x = -8 - y

(-8 - y)y = -9

-y² - 8y + 9 = 0

y² + 8y -9 = 0

Solve for y by factoring:

(y - 1)(y + 9) = 0

y = 1, -9

x(1) = -9

x = -9

x(-9) = -9

x = -9/-9 = 1

The value of x in the equation y = 9 sec(2x) at [0, π/4) ∪ (π/4, π/2] is x = 1/2[sec₋¹(y/9)]

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

y = 9 sec(2x)

The interval of x is also given as

[0, π/4) ∪ (π/4, π/2]

This means that the values of x is from 0 to π/2, however, the function is undefined at x = π/4 i.e. there is a hole at x = π/4

Next, we set the equation to y

So, we have

9 sec(2x) = y

Divide both sides by 9

sec(2x) = y/9

Take the arc sec of both sides of the equation

2x = sec₋¹(y/9)

Divide both sides by 9

x = 1/2[sec₋¹(y/9)]

Hence, the value of x is x = 1/2[sec₋¹(y/9)]

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The exact values are;

sin⁡θ = -5/√29

cos⁡θ = 2/√29

sec⁡θ =√29/2

csc⁡θ = -√29/5

cot⁡θ = -2/5

We are given that

tan⁡θ=-5/2 and sin⁡θ>0

First quadrant: I, 0°<θ<90°

We can find the first quadrant between 0° and 90° , the values from 0 to the positive numbers for the x-axis and the y-axis, the functions sine, cosine and tangent will always have positive values.

sin⁡θ = -5/√29

cos⁡θ = 2/√29

sec⁡θ =√29/2

csc⁡θ = -√29/5

cot⁡θ = -2/5

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Answer:

-[tex]\sqrt{3}[/tex]/2

Step-by-step explanation:

cos is negative in quad II

cos(210)= -cos(30) = -[tex]\sqrt{3}[/tex]/2

Answer:

24

Step-by-step explanation:

14 + 17 + 21 + 28 +40 / 5

120 / 5

24

The decimal 15.75 into a percentage is 1575%

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

Decimal = 15.75

This means that

Number = 15.75

Multiply the number by 1

so, we have the following representation

Number = 15.75 * 1

Express 1 as 100%

This gives

Number = 15.75 * 100%

Evaluate the products of 15.75 and 100

So, we have

Number = 1575%

Hence, the percentage is 1575%

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Answer:

Step-by-step explanation:

To calculate the mean (average) weekly earnings of workers in the occupations listed for 2010, we need to add up the earnings for each occupation and divide by the total number of occupations.

Forestry, logging and support: $971

Manufacturing: $977

Transportation and warehousing: $900

Construction: $1071

Retail trade: $501

Total earnings: $4,420

Total number of occupations: 5

Mean weekly earnings: $4,420 ÷ 5 = $884

Therefore, the mean (average) weekly earnings of workers in the occupations listed for 2010 is $884.

The probability of drawing a second marble that is blue is 3/5

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

A red marble is drawn from a bag containing 3 red and 3 blue marbles.

If the marbles were not replaced, then we have

P(Red) = 3/6

Now there are

3 blue marbles and 2 red marbles left

So, we have

The probability of choosing a blue marble, after a red marble is

P(Blue) = 3/5

Evaluate

P(Blue) = 3/5

Hence, the probability of choosing a blue marble, after a red marble is 3/5

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If Yogi Charges 4.30$ per square foot for the carpet plus $75 installation fee, the total cost of installing carpet in the hotel lobby with Yogi is $2,225.

To determine the total cost of installing carpet in the hotel lobby with Yogi, we need to know the area of the lobby in square feet. Once we have the area, we can use Yogi's pricing scheme to calculate the total cost.

Assuming that we have measured the area of the lobby to be 500 square feet, we can calculate the total cost as follows:

Cost of carpet = area × price per square foot

= 500 × $4.30

= $2,150

Cost of installation = $75

Total cost = cost of carpet + cost of installation

= $2,150 + $75

= $2,225

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Step-by-step explanation:

[tex] \frac{ {x}^{2} + 3x - 10 }{x + 5} \\ = \frac{(x + 5)(x - 2)}{(x + 5)} \\ = (x - 2)[/tex]

The probability that the woman's height is between 62.9 in and 63.9 in is 0.1580

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

So, the z-scores are

z = (62.9 - 63.6)/2.5 = -0.28

z = (63.9 - 63.6)/2.5 = 0.12

i.e. between a z-score of -0.28 and a z-score of 0.12

This is represented as

Probability = (-0.28 < z < 0.12)

Using a graphing calculator, we have

Probability =  0.1580

Hence, the probability is 0.1580

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Answer:

Less than 180

Step-by-step explanation:

Minor: Less than 180

Major: More than 180

Answer:

We can simplify the expression as follows:

3(-4b) - 2(a - b - c) = -12b - 2a + 2b + 2c

Combining like terms, we get:

= -2a - 10b + 2c

Therefore, the expression 3(-4b) - 2(a - b - c) is equal to option (2), -2a - 10b + 2c.

Step-by-step explanation:

Im smart

The probability of getting blue or green is 1/3, the probability of getting blue is 3/4 and there are 8 blue marbles.

a) If the probability of getting a yellow marble is 2/3, then the probability of getting a blue or green marble is:

P(blue or green) = 1 - P(yellow) = 1 - 2/3 = 1/3

b) If the probability of getting a green marble is 1/4, then the probability of getting a blue marble is:

P(blue) = 1 - P(green) = 1 - 1/4 = 3/4

c) The total number of marbles in the bag is 24. Let x be the number of blue marbles. Then the number of yellow marbles is 2x, and the number of green marbles is 24 - x - 2x = 24 - 3x. We know that:

2x/24 = 2/3 (probability of getting a yellow marble is 2/3)

=> x = 8

Therefore, there are 8 blue marbles in the bag.

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Answer:

[tex]sec(-\theta)=-3.1[/tex]

[tex]sin(-\theta)=-0.62[/tex]

Step-by-step explanation:

Part a.

[tex]sec(\theta)[/tex] is related to [tex]cos(\theta)[/tex] through a reciprocal relationship [tex]sec(\theta)=\frac{1}{cos(\theta)}[/tex].

Since [tex]cos(\theta)[/tex] is an even function (reflecting left-right across the y-axis doesn't change the graph), then [tex]sec(\theta)[/tex] is also an even function.

For any input in the domain, even functions produce the same output if the opposite of the input is used.  In other words, if "f" is an even function, for all x in the domain of f, [tex]f(x)=f(-x)[/tex].

Thus, for the secant function, if a mystery value "x" is used as an input, and -3.1 is obtained as an output, then if the opposite of x, or -x, is input into the secant function, the output will also be -3.1.

[tex]sec(-\theta)=-3.1[/tex]

Part b.

The [tex]sin(\theta)[/tex] function does not reflect left-right across the y-axis to produce the same graph, so the sine function is not even.

However, the sine function can be rotated 180 degree about the origin, sometimes thought of as reflecting through the origin, to produce the same graph.  Visually, this is an "odd" function.

For any input in the domain, if the opposite of the input is used, odd functions produce the opposite of the original output.  In other words, if "g" is an odd function, for all x in the domain of g, [tex]-g(x)=g(-x)[/tex].

Thus, for the sine function, if a mystery value "x" is used as an input, and 0.62 is obtained as an output, then if the opposite of x, or -x, is input into the sine function, the output will be the opposite of 0.62, meaning -0.62.

[tex]sin(-\theta)=-0.62[/tex]

Answer:380.1 square inches

Step-by-step explanation:Area of a circle in terms of diameter: Area = π· (d2) 2 = 3.14· (222) 2 = 3.14· (11) 2 = 380.1 square inches (*)

The given function does not represent the curve passing through the given points. The multiplicative rate of change of the function is 5.

A curve is a continuous, smooth line that gradually alters direction. Rather than being straight line that follows a curve may be referred to as a curve line. A curve line can be made by connecting a number of non-straight-line-lying sites. In mathematics, a curve is a geometrical object that can be described by equations, such as parametric or function equations.

The rate of change of a function is the ratio of the change in the output (y) to the change in the input (x). In this case, we can calculate the rate of change between the first two points:

Changing at what rate between (1, 10) and (2, 50)?

Y change = 50 – 10 = 40

Variation in x = 2 - 1 = 1

Rate of change equals change in y/change in x, or 40/1, or 40.

In a similar manner, we can determine how quickly the second and third points will change:

The change between (2, 50) and (3, 250) is as follows:

Changing y by 250 - 50 equals 200

Variation in x = 3 - 2 = 1

Rate of change equals change in y/change in x, which is 200/1, or 200.

Now that we have the second rate of change, we can compute the multiplicative rate of change by dividing it by the first:

Multiplicative rate of change is equal to 200/40, or 5.

The function's multiplicative rate of change is thus 5.

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Answer:

Step-by-step explanation:

well if you know the term than you know the pattern

Answer:

We can use synthetic division to divide the polynomial 2x^3 + 522x^2 - 11x + 104 by x + 8, since x + 8 is a factor of the polynomial:

-8 | 2   522   -11   104

  |     -16   -412  2584

  |--------------------

  2   506   -423  2688

The numbers in the bottom row of the synthetic division represent the coefficients of the quotient polynomial, in order of decreasing degree. So the quotient polynomial is:

2x^2 + 506x - 423

Therefore, the other factor of the polynomial is 2x^2 + 506x - 423.

The exponential growth model is represented by the function y = 10 · 3ˣ. (Correct choice: D)

In this problem we must determine what function does represent an exponential growth model. With this purpose, we must define and understand the following functions:

Exponential growth model

y = a · rˣ, for r > 1.

Exponential decay model

y = a · rˣ, for 0 < r < 1.

Polynomic model

y = ∑ cₙ · xⁿ

The function y = 10 · 3ˣ represents an exponential growth model.

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If x + 5 is a factor of the given polynomial, then (x + 5) must divide the polynomial evenly, meaning that the remainder is 0 when the polynomial is divided by x + 5.

We can use polynomial long division or synthetic division to find the quotient and remainder, but it's easier to use the fact that if x + 5 is a factor, then (-5) must be a root of the polynomial.

So, we can substitute x = -5 into the polynomial and set it equal to 0 to find m:

-3(-5)^4 - 10(-5)^3 + 20(-5)^2 - 22(-5) + m = 0

Simplifying and solving for m:

-3(625) + 10(125) + 20(25) + 110 + m = 0

-1875 + 1250 + 500 + 110 + m = 0

m = 1015

Therefore, m = 1015 so that x + 5 is a factor of - 3x^4 - 10x^3 + 20x^2 - 22x + m.

The speed in miles per hour is 111.2 and the direction angle in degrees is 260.2.

We are given the speed of a plane on a bearing of N [tex]75^\circ[/tex] E and its ground speed. We have to find its speed in miles per hour and the direction angle in degrees. We will apply the formula of projection for both the x-axis and y-axis.

As we know, projection, R = V + W

Now, the x-axis projection will be R cos[tex]15^\circ[/tex] according to the angle given to us. Therefore, R cos [tex]15^\circ[/tex] = V cos[tex]30^\circ[/tex] + [tex]W_{x}[/tex]

The y-axis projection,

R sin [tex]15^\circ[/tex] = V sin [tex]30^\circ[/tex] + [tex]W_{y}[/tex]

From here, now we will find [tex]W_{x}{[/tex] and [tex]W_{y}[/tex]

[tex]W_{x}{[/tex] = 330 cos[tex]15^\circ[/tex] - 390 cos[tex]30^\circ[/tex]

[tex]W_{x}[/tex] = -19 miles/hour

[tex]W_{y}[/tex] = 330 sin[tex]15^\circ[/tex] - 390 sin[tex]30^\circ[/tex]

[tex]W_{y}{[/tex] = -109.6 miles per hour

Now, W = [tex]\sqrt{(W_{x})^{2} + (W_{y})^{2{}}[/tex]

W = [tex]\sqrt{(-19.0)^{2} + (-109.6})^{2{}}[/tex]

W = 111.2 miles/hour

Now, we will find the angle with the help of tan θ.

tan θ = [tex]\frac{W_{y}}{W_{x}}[/tex]

tan θ = [tex]\frac{-109.6}{-19.0}[/tex]

θ =  [tex]tan ^{-1} (\frac{109.6}{19.0})[/tex]

θ = 260.[tex]2^\circ[/tex]

Therefore, the speed in miles per hour is 111.2 and the direction angle in degrees is 260.2.

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If the line passes through the point (2,8) that cuts off the least area from the first quadrant, the slope is 8/3 and the y-intercept is 0.

To find the equation of the line that passes through the point (2, 8) and cuts off the least area from the first quadrant, we need to first determine the slope of the line. The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept.

We can use the point-slope form of the equation of a line to find the slope. The point-slope form is y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line. Plugging in (2, 8) as the point, we get:

y - 8 = m(x - 2)

Next, we want to minimize the area that the line cuts off in the first quadrant. Since the line passes through the origin (0, 0), the area cut off by the line in the first quadrant is equal to the product of the x- and y-intercepts of the line.

We can express the x-intercept in terms of y by setting y = 0 in the equation of the line and solving for x:

0 - 8 = m(x - 2)

x = 2 + 8/m

The y-intercept is simply the y-coordinate of the point where the line intersects the y-axis, which is given by:

y = mx + b

8 = 2m + b

b = 8 - 2m

We can now express the area cut off by the line as:

A = x*y

A = (2 + 8/m)*8 - (8 - 2m)*2/m

A = (16 + 64/m) - (16 - 4m)/m

A = 64/m + 4m/m

To minimize the area, we can take the derivative of A with respect to m and set it equal to zero:

dA/dm = -64/m² + 4/m² = 0

64 = 4

m = 8

Plugging m = 8 into the equation for the x-intercept, we get:

x = 2 + 8/8 = 3

So the equation of the line is y = 8x/3.

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Answer:

Based on the hypothesis test conducted with a significance level of 5%, we fail to reject the null hypothesis that the proportion of smokers in the city is 20%. This means that we do not have sufficient evidence to conclude that the proportion of smokers is less than 20%. The p-value of 0.828 suggests that there is a high probability that the observed proportion of smokers in the sample is due to chance and not a true difference in the proportion of smokers in the population. Therefore, we cannot conclude that the city has a lower proportion of smokers than 20%.

In this hypothesis test set up by the researcher, the p-value is 0.828, which is greater than the significance level (0.05). Therefore, we do not reject the null hypothesis, meaning there is not enough statistical evidence to validate the researcher's claim that the proportion of smokers is less than 20%

A hypothesis test in statistics uses test statistics based on sample data to accept or reject a null hypothesis. In this scenario, the null hypothesis (H0) states that the proportion of smokers (p) is 20%. The alternative hypothesis (Ha) claims that the proportion of smokers is less than 20%. The p-value is a measure of the probability that the observed data could occur under the null hypothesis. In our case, a p-value of 0.828 means that there is an 82.8% chance of observing the data if the true proportion of smokers is 20%, or higher.

Usually a threshold known as the significance level (in this case 5% or 0.05) is used to determine whether the null hypothesis should be rejected or not. If the p-value is less than or equal to the significance level, it suggests that the observed data is inconsistent with the null hypothesis, and the null is usually rejected. However, since our p-value is greater (0.828 > 0.05), we would not reject the null hypothesis, suggesting that there is not enough evidence to support the researcher's claim that the proportion of smokers is less than 20%.

Therefore, the conclusion is that the researcher's claim cannot be validated using the provided data.

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The vector v can be expressed as v = -0.161 i + 0.618 j, where i and j are the unit vectors

To express a vector v in terms of i and j, we need to find its x and y components. The magnitude ||v|| of a vector is given by:

||v|| = √(v₁² + v₂²)

where v₁ and v₂ are the x and y components of v, respectively.

The direction angle θ of a vector with respect to the positive x-axis is given by:

θ = atan(v₂/v₁)

where atan denotes the arctangent function.

In this problem, we are given that the magnitude ||v|| of the vector v is 2/3, and its direction angle θ with respect to the positive x-axis is 116 degrees. Therefore, we can write:

||v|| = √(v₁² + v₂²) = 2/3

θ = atan(v₂/v₁) = 116°

Solving for the x and y components, we get:

v₁ = ||v|| cos(θ) = (2/3) cos(116°) ≈ -0.161

v₂ = ||v|| sin(θ) = (2/3) sin(116°) ≈ 0.618

Therefore, the vector v can be expressed as:

v = -0.161 i + 0.618 j

where i and j are the unit vectors in the x and y directions, respectively.

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Answer: 12

Step-by-step explanation:

We have to find h so the first step is to subtract the 11 from the 20

3/4h=9

Then to move the 3/4 over to the other side, you must multiply it by it's reciprocal, 4/3.

h= 4/3 x 9

This equals 12.