Either show that the following sets of vectors are linearly independent, or find a linear relation between them (the T means transpose): (a) x^(1) = [ 1 1 0 ]^T , X^(2) = [ 0 1 1 ]^T , X^(3) = [1 0 1 ]^T. (b) x^(1) = [ 2 1 0 ]^T , X^(2) = [ 0 1 0 ]^T , X^(3) = [-1 2 0 ]^T.

The characteristics of each function has been explored based on the table below.

The graph of each function is shown in the image attached below.

In order to use the given functions to complete the table, we would have to substitute each of the values of x (x-values) into the function and then evaluate as follows;

A. f(x) = 2x

x                      process             f(x)

-2                  f(-2) = 2(-2)            -4

-1                   f(-1) = 2(-1)              -2

0                   f(0) = 2(0)               0

1                    f(1) = 2(1)                 2

2                   f(2) = 2(2)                4

B. G(x) = x²

x                      process             f(x)

-2                  f(-2) = (-2)²            4

-1                   f(-1) = (-1)²              1

0                   f(0) = (0)²               0

1                    f(1) = (1)²                 1

2                   f(2) = (2)²               4

B. H(x) = 2ˣ

x                      process             f(x)

-2                  f(-2) = (2)⁻²            1/4

-1                   f(-1) = (2)⁻¹              1/2

0                   f(0) = (2)⁰                1

1                    f(1) = (2)¹                 2

2                   f(2) = (2)²               4

In this scenario and exercise, we would use an online graphing calculator to plot each of the given functions as shown in the graph attached below.

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The rate at which the surface area of the sphere is decreasing when the radius is 8 meters and decreasing at a rate of 3 m/s is approximately -192π square meters per second.

Let's start by finding the formula for the surface area of a sphere. The surface area (A) of a sphere with radius (r) is given by the formula:

A = 4πr²

Now, we need to find the rate at which the surface area is changing with respect to time (t) when the radius is 8 meters and the rate of change of radius is 3 m/s.

To find dA/dt, we need to differentiate the formula for surface area with respect to time, using the chain rule:

dA/dt = d/dt (4πr²) = 8πr (dr/dt)

Here, we have used the fact that the derivative of r² with respect to time is 2r (dr/dt) by the chain rule. Now, we can substitute the given values into the formula to find the rate of change of surface area:

r = 8 m (given)

dr/dt = -3 m/s (negative sign because the radius is decreasing)

π ≈ 3.14 (constant)

dA/dt = 8πr (dr/dt)

dA/dt = 8π(8)(-3)

dA/dt = -192π

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The second fuel line must be able to fill the oil tanker at a rate of 2.67 tanks per hour in order to fill the tanker in 45 minutes when used together with the first fuel line.

Rate of the first fuel line = 1 tank / 3 hours = 1/3 tanks per hour

Let's assume that the rate of the second fuel line is x tanks per hour. When both fuel lines are used together, their combined rate is the sum of their individual rates:

Combined rate = Rate of the first fuel line + Rate of the second fuel line

1 tank / 0.75 hours = (1/3) tank per hour + x tanks per hour

Simplifying the equation, we get:

1 tank / 0.75 hours - 1/3 tank per hour = x tanks per hour

x = (1 tank / 0.75 hours - 1/3 tank per hour)

x = 2.67 tanks per hour

Therefore, the second fuel line must be able to fill the oil tanker at a rate of 2.67 tanks per hour in order to fill the tanker in 45 minutes when used together with the first fuel line.

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Answer: 131.95 cm

Step-by-step explanation:

The diameter of the semicircle is 42 cm, which means the radius is half of that:

r = d/2 = 42/2 = 21 cm

The circumference of a full circle with radius "r" is given by the formula:

C = 2πr

Since this is a semicircle, we need to divide the circumference by 2:

C/2 = πr

Substituting the value of "r" we found above, we get:

C/2 = π(21)

Simplifying:

C/2 = 21π

C = 2(21π)

C ≈ 131.95

Rounding to the nearest hundredth, we get:

C ≈ 131.95 cm

Therefore, the circumference of the semicircle is approximately 131.95 cm.

Answer:

  5240 kg/m³

Step-by-step explanation:

You want the average density of a planet with radius 6052 km and mass 4.87×10^27 g.

The mass is given in grams, and the corresponding unit in the desired answer is kilograms. There are 1000 g in 1 kg, so 4.87×10^27 g = 4.87×10^24 kg.

The radius is given in km, and the corresponding unit in the desired answer is meters. There are 1000 meters in 1 km, so 6052 km = 6052×10^3 m. (We could adjust the decimal point, but we choose to let the calculator do that.)

The units of density tell you it is computed by dividing the mass by the volume:

  ρ = mass/volume

The volume of the sphere is found using the given formula, so the density is ...

  ρ = (4.87×10^24 kg)/(4/3π(6052×10^3 m)^3)

  ρ ≈ 5240 kg/m³

The average density of the planet is about 5240 kg/m³.

__

Additional comment

This is comparable to the average density of Earth, which is about 5520 kg/m³.

The probability of selecting a ball that has an even number from a bag containing 13 balls numbered 1 through 13 is 6/13 or approximately 0.4615. This is because out of the 13 balls, 6 of them have even numbers.

To calculate the probability, we use the formula:

Probability = Number of Favorable Outcomes / Total Number of Outcomes

In this case, the favorable outcomes would be the number of even numbers, which is 6. The total number of outcomes would be 13 (the total number of balls in the bag). Therefore, the probability of selecting a ball that has an even number is 6/13.

Another way to look at this is to think of the probability as the ratio of favorable outcomes to total possible outcomes. In this case, the ratio is 6:13, which can be simplified to 3:6 or 1:2. Since one out of every two outcomes is favorable, the probability is 0.5, which is the same as 6/13.

To further understand the concept of probability, consider the example of a fair coin. If a fair coin is flipped, the probability of getting either heads or tails is 50%. This is because the ratio of favorable outcomes (one head and one tail) to total possible outcomes (two heads and two tails) is 1:2.

In conclusion, the probability of selecting a ball that has an even number from a bag containing 13 balls numbered 1 through 13 is 6/13 or approximately 0.4615.

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The formula for percent from the board can be expressed this way:

The total number of students that like pineapple pizza   * 100.

      The total number of students

A (i). The part is the number of students that like pineapple pizza which is 5.

(ii). The whole is the total number of students namely; 7 + 12+ 5 + 3 + 6 = 33.

(iii). Plugging the part and whole into the equation, we will have: 5/33 * 100

(iv). Final answer is = 15.15%

A percentage is a mathematical expression that is aimed at calculating the proportion of a given quantity out of 100. In the question, we are requested to find what proportion of students like pineapple pizza.

To get the answer, we first have to determine the number of students that like pineapple pizza which is 5 according to the table. Next, we get the sum of students which is 33. The percentage will then get the fraction and multiply the result by 100. The final answer is 15.15%.

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Based on the past week's sales data, the baker can expect to sell Option C 220 caramel, 130 gingerbread, and 420 mint choco chip cookies.

The population in statistics refers to the total set of people or things that we are attempting to analyse and make conclusions about. Typically, this group is too big to see or measure directly, so we use statistical methods to infer some of its features from a smaller collection of people or things, known as a sample. A representative portion of the population was selected for the sample in order to learn more about the population as a whole. The purpose of statistical analysis is to draw conclusions about the population using data from the sample.

The number of weeks in a month= 4.

Using the given table the estimated number of sales for the month are:

Salted caramel: 51 x 4 = 204

Gingerbread: 31 x 4 = 124

Mint: 99 x 4 = 396

Hence, based on the past week's sales data, the baker can expect to sell Option C 220 caramel, 130 gingerbread, and 420 mint choco chip cookies.

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The three ordered pairs that are most likely the added data points are:

B: (4,10), (4,11), (4,12)

C: (8,9), (8,10), (8,11)

D: (10,10), (10,11), (10,12)

In order to determine the added data points, we first need to note that the data points have a slightly linear pattern with a negative slope. Also, a linear regression on the points will give us:  linear regression on the given data points, we obtain the equation: y = -1.2x + 10.8

In selecting the three added data points, a good criterion will be for the points to be as close as possible to the original line of best fit. The data points in option A are close to slope 1 so adding them will significantly shift the line of best fit. The data points in options B, C, and D are in a vertical line and will not shift the line of best fit. So, they are most likely the best data points to add.

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The equation in slope-intercept form that describes the relationship in the graph is: y = -20x + 100.

An equation is a mathematical statement that shows the equality between two expressions. It contains one or more variables, and the values of these variables can be solved for to make the equation true. An equation can be represented using symbols, numbers, and operators such as addition, subtraction, multiplication, division, and exponents. Equations are used in many fields of mathematics and science, and they are a fundamental tool for solving problems and making predictions.

Here,

Part A:

The equation in slope-intercept form that describes the relationship in the graph is:

y = -20x + 100

Part B:

To determine the equation, we need to find the slope of the line passing through the two given points.

Slope, m = (change in y) / (change in x)

m = (0 - 100) / (5 - 0)

m = -20

The y-intercept can be found by substituting the slope and any point on the line into the slope-intercept form of the equation:

y = mx + b

100 = (-20)(0) + b

b = 100

Therefore, the equation in slope-intercept form is:

y = -20x + 100

Part C:

In the given situation, the slope represents the rate of decrease in the percentage of battery remaining per hour. The negative value indicates that the percentage of battery remaining decreases over time, and the magnitude of the slope (-20) indicates that the battery percentage decreases by 20% per hour.

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The equation of an exponential decay function with a range of (-∞,10), vertical shrink of 3/4, and vertical shift up 10 units is f(x) = 10 - (3/4)^x.

An exponential decay function can be written as f(x) = ab^x where b is the base of the exponential function and a is the initial value. If b is between 0 and 1, the function will exhibit exponential decay. Since we need a vertical shrink of 3/4, we can set b = 3/4.

We also need a vertical shift of 10 units, so we add 10 to the function. The resulting function is: f(x) = 10 - (3/4)^x. Therefore, the equation of an exponential decay function with a range of (-∞,10), vertical shrink of 3/4, and vertical shift up 10 units is f(x) = 10 - (3/4)^x.

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There are 300 ways a person can go from P to Q and return by a different bus.

To find the number of ways a person can go from P to Q and return by a different bus, we can use the combination formula:

nCr = n! / (r! × (n-r)!)

where n is the total number of options, and r is the number of choices we want to make.

In this case, there are 25 buses running between P and Q, so the total number of options for going from P to Q and returning by a different bus is 25 × 24 (since we have 25 choices for the first leg of the journey, and 24 choices for the second leg).

To find the total number of ways to make this choice, we can use the combination formula with n = 25×24 and r = 2

nCr = (25×24)! / (2! × (25×24-2)!)

= (25 × 24)! / (2! × 23!)

= (25×24) / 2

= 300

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Therefore, the observed value of the test statistic for testing the null hypothesis H0: p = 0.5 versus the alternative hypothesis H1: [tex]p > 0.5[/tex]is z= 1.8. The correct answer is (e) z=[tex]0.59— 0.5/√0.5(0.5)/100.[/tex]

In this particular question, the observed value of the test statistic for testing the null hypothesis H0: p = 0.5 versus the alternative hypothesis H1: p > 0.5 is to be determined. The question provides a random sample of 100 likely voters in a small city, which produced 59 voters in favor of Candidate A. The formula for calculating the test statistic is as follows:

z= (p- P) / sqrt[P(1-P) / n]

where p is the sample proportion, P is the hypothesized population proportion, and n is the sample size.

Substituting the given values into the formula, we get:

z= [tex](0.59- 0.5) / sqrt[0.5(0.5) / 100][/tex]

z= [tex]0.09 / 0.05[/tex]

z=[tex]1.8[/tex]

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Answer:

14 = 2 x 7

28 = 2² x 7

56 = 2³ x 7

Answer: 65

Step-by-step explanation:

The estimated joint probability for the "low income and 21-35 age group" cell is 0.108.

In the table given, individuals are cross-classified by their income level and age group. We have to find the estimated joint probability for the "low income and 21-35 age group" cell. The joint probability of the low income and 21-35 age group is calculated as follows;

From the given table, the number of people in the "low income and 21-35 age group" cell is 120.The total number of people in the sample = 120 + 100 + 75 + 150 + 160 + 100 + 160 + 180 + 160 = 1105.

The estimated joint probability for the "low income and 21-35 age group" cell is:P(Low income, 21-35 age group) = 120/1105P(Low income, 21-35 age group) = 0.108. Therefore, the estimated joint probability for the "low income and 21-35 age group" cell is 0.108.

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Answer/Step-by-step explanation:

We know that sin 60° = √3/2, and we can use this information to find cos 60°:

cos 60° = √(1 - sin^2 60°) = √(1 - 3/4) = √(1/4) = 1/2.

To find the measure of angle 3, we can use the formula:

cos(180° - 2x) = -cos(2x)

We want to solve for x when cos 2x = √(√3/2) = √3/2 * 1/√2 = √3/2√2

cos(180° - 2x) = -cos(2x)

cos(2x) = √3/2√2

180° - 2x = 150° or 210°

x = 15° or 35°

Therefore, the measure of angle 3 can be either 15° or 35°.

I hoped this helped!  

: )

Step-by-step explanation:

originally - (6,-1)     After translation - (1,4)

originally - (8,-1)     After translation - (3,4)

originally - (4,-7)     After translation - (-1,-2)

originally - (7,-9)     After translation - (3,-4)

originally - (9,-9)     After translation - (4,-4)

Therefore , the solution of the given problem of unitary method comes out to be the parking lot has 60 trucks.

It is possible to accomplish the objective by using this widespread convenience, pre-existing variables, or all significant components from the initial Diocesan adaptable survey that adhered to a specific methodology. If it doesn't, both of the essential components of a term confirmation outcome will undoubtedly be lost, but if it is, there is going to be another opportunity to contact the entity.

Here,

Let's begin by giving the unknowns variables:

x = number of vehicles

Van count = 4 times

Since there are 205 cars overall, the number of lorries is equal to (205 - 11x).

The value of x can be determined using the second bit of knowledge:

=> 4x/3 = (205 - 11x)/2

When we simplify this solution, we obtain:

=> 8x = 3(205 - 11x)

=> 8x = 615 - 33x

=> 41x = 615

=> x = 15

We can now determine the quantity of vans:

Van count = 4x = 4(15) = 60

Consequently, the parking lot has 60 trucks.

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Answer: 40/7 or 5.7

Step-by-step explanation:

using angle bisector theorem we get,

x/5 = 8/7

Upon substituting our given values in above equation, we will get:

x/5 X 5 = 8/7 X 5

x = 40/7 or 5.7

Answer:

3.3

Step-by-step explanation:

Angle bisector theorem: In a triangle, the angle bisector of any angle will divide the opposite side in the ratio of the sides containing the angle.

       [tex]\dfrac{x}{8-x}=\dfrac{5}{7}[/tex]

Cross multiply,

    x *7 = 5*(8 - x)

        7x = 40 - 5x

   7x + 5x = 40

        12x   = 40

             [tex]x = \dfrac{40}{12}\\\\x = 3.3[/tex]

Answer: Jills fourth number is 28

Step-by-step explanation:

Answer:

4 for up but of R it is 10

Step-by-step explanation:

378/92 equals 4 but 10 is the remainder

Answer:

8%

Step-by-step explanation:

[tex]= .08[/tex]

This is the answer as a decimal. To change a decimal to a perfect multiply the decimal by 100%

[tex].08 \times 100\% = 8\%[/tex]

Answer:

linear

Step-by-step explanation:

gradient 1 = [tex]\frac{-3-12}{6-3}[/tex] = -5

gradient 2 = [tex]\frac{-18+3}{9-6}[/tex] = -5

therefore, the graph is linear as the gradient is constant

Answer:

The equation of a vertical line in the graph, which is parallel to y-axis is x = a. The slope of a vertical line is infinity or undefined as it has no y-intercept and the denominator in the slope formula is zero.

Step-by-step explanation:

When we write out the system of equations, we will get:

[tex]x+y=18[/tex]

[tex]x-y=-8[/tex]

We can then use elimination to get just one variable.

[tex]x+y=18[/tex]

[tex]x-y= -8[/tex]

Which gives us: [tex]2x=10[/tex]

We then divide by 2 and get: [tex]x=5[/tex]

We next plug that into one of the two equations and solve for the other variable: [tex](5)+y=18[/tex]

We subtract 5 from both sides and get: [tex]y=13[/tex]

So the two numbers are 13 and 5.

Answer:

There was a 33.3% decrease in length.

Step-by-step explanation:

4/12 = 0.33333333

0.333333*100 = 33.3%

Therefore, the order of the samples from smallest to largest amount (moles) of OY is: Go, Y, Na, Mn.

In order to list the samples in order of the amount (moles) from smallest to largest OY, we need to calculate the number of moles of each element. Let's start with the given samples:Na -[tex]0.10 gY - 0.10 gGo - 0.10 gMn - 0.10 g[/tex]

Now, let's find the number of moles of each element:Na: The molar mass of Na is 22.99 g/mol.Number of moles of Na = mass of Na/molar mass of Na= 0.10 g/22.99 g/mol= 0.00435 molY: The molar mass of Y is 88.91 g/mol.Number of moles of Y = mass of Y/molar mass of Y= 0.10 g/88.91 g/mol= 0.00112 molGo: The molar mass of Go is 118.71 g/mol.Number of moles of Go = mass of Go/molar mass of Go= 0.10 g/118.71 g/mol= 0.00084 molMn: The molar mass of Mn is 54.94 g/mol.

Number of moles of Mn = mass of Mn/molar mass of Mn= 0.10 g/54.94 g/mol= 0.00182 molNow, we can list the samples in order of the amount (moles) from smallest to largest OY:Go (0.00084 mol) < Y (0.00112 mol) < Na (0.00435 mol) < Mn (0.00182 mol)

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Answer:

The volume is 1474.45 mm^2

Step-by-step explanation:

First, you have to find the radius. To do this, you divide the diameter by 2. This means that the radius is 8.

Then you use the formula V=3.14*r^2*(h/3)

V=3.14*8^2*(22/3)=1474.45 mm^2