what is a random variable? group of answer choices the outcome of a probability experiment is often a count or a measure. when this occurs, the outcome is called a random variable.

The points A, B and C are collinear.​ because they have the same slope

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

A(-3;-5), B(0;-3) and C(6;1),

To show that the points A, B and C are collinear, we can use the slope formula

The slope is calculated as

Slope = Change in y/Change in x

Using the above as a guide, we have the following:

Slope AB = (-3 + 5)/(0 + 3) = 2/3

Slope CB = (-3 - 1)/(0 - 6) = 2/3

See that the calculated slopes above have the same values

This means that the points A, B and C are collinear.​

Read more about linear relation at

https://brainly.com/question/30318449

#SPJ1

She needs to make about $ 17.2, so the correct option is the last one.

First, we need to add all the values in the table to see how much she needs, we will get:

A = 653 + 245 + 299 + 462 + 435 + 136 + 512 = 2,747

Now, we know that Dora works 40 hours per week, 4 weeks a month, so she works a total of.

40*4 hours = 160 hours.

Then the amount that she needs to make per hour is given by the quotient between the total amount she needs and the number of hours:

N =  2,747/160

N = 17.2

The only option that has that in the range is the last one.

Learn more about quotients at:

https://brainly.com/question/629998

#SPJ1

Picard's theorem states that a first-order ordinary differential equation of the form y' = f(x, y) with an initial condition y(x0) = y0 has a unique solution if the function f(x, y) is continuous and satisfies the Lipschitz condition with respect to y in some region containing the initial point (x0, y0).

For the given differential equations, we can apply Picard's theorem to determine whether they have unique solutions.

(i) The function f(x, y) = t cos^-1(y) is continuous and satisfies the Lipschitz condition with respect to y in some neighborhood of (0, 1), so by Picard's theorem, the equation has a unique solution.

(ii) The function f(x, y) = tx is continuous and satisfies the Lipschitz condition with respect to y in some neighborhood of (0, y0), so by Picard's theorem, the equation has a unique solution.

(iii) The function f(x, y) = 1/(t^2*y^2) is not continuous at (0, 0), so we cannot apply Picard's theorem to determine whether the equation has a unique solution.

(iv) The function f(x, y) = y^(2/3) is continuous and satisfies the Lipschitz condition with respect to y in some neighborhood of (0, 0), so by Picard's theorem, the equation has a unique solution.

Learn more about Picard's theorem here: brainly.com/question/30906464

#SPJ11

To prove the equation [tex]\frac{1}{\tan(2\theta) - \tan(\theta)} - \frac{1}{\cot(2\theta) - \cot(\theta)} = \cot(\theta)[/tex] we'll simplify the left side, this is shown below:

Using the trigonometric identities [tex]\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}[/tex] and [tex]\cot(\theta) = \frac{1}{\tan(\theta)}[/tex]

We can rewrite the expression as [tex]\frac{1}{\tan(\theta)(1 - \tan(\theta))} - \frac{\tan(\theta)}{\tan(\theta)(1 - \tan^2(\theta))}[/tex]

Combining the fractions with a common denominator, we obtain [tex]\frac{1 - \tan(\theta)}{\tan(\theta)(1 - \tan(\theta))}[/tex]

Simplifying further, we cancel out the [tex](1 - tan(\theta))[/tex] terms, leaving us with [tex]\frac{1}{\tan(\theta)}[/tex] = [tex]\cot(\theta)[/tex], which is equivalent to the right side of the equation.

Thus, the equation is proven.

Read more about trigonometric identities here:

https://brainly.com/question/7331447
#SPJ1

We can see that the pattern progresses by adding 3 to the previous perfect square to obtain the next term. The next few terms of the sequence would be:

25 (16 + 3)

36 (25 + 3)

49 (36 + 3)

64 (49 + 3)

81 (64 + 3)

...

We can continue this pattern to find as many terms as desired.

What is Number Sequences?

In mathematics, a number sequence is an ordered list of numbers that follow a specific pattern or rule. Each number in the sequence is called a term, and the position of a term in the sequence is called its index.

The given pattern appears to be a sequence of perfect squares starting from 1 and increasing by 3 at each step. We can verify this by observing that:

The first term is 1 which is a perfect square.

The second term is 4 which is a perfect square and is obtained by adding 3 to the previous term 1.

The third term is 9 which is a perfect square and is obtained by adding 3 to the previous term 4.

The fourth term is 16 which is a perfect square and is obtained by adding 3 to the previous term 9.

Therefore, we can see that the pattern progresses by adding 3 to the previous perfect square to obtain the next term. The next few terms of the sequence would be:

25 (16 + 3)

36 (25 + 3)

49 (36 + 3)

64 (49 + 3)

81 (64 + 3)

...

We can continue this pattern to find as many terms as desired.

To learn more about number sequence from the given link:

https://brainly.com/question/7043242

#SPJ4

Based on the graph, the quadratic equation is y = (x - 2)² - 9.

In Mathematics, the vertex form of a quadratic function is represented by the following mathematical equation:

f(x) = a(x - h)² + k

Where:

Based on the information provided about the vertex (2, -9) and the other points (5, 0), we can determine the value of "a" as follows:

y = a(x - h)² + k

0 = a(5 - 2)²  - 9

9 = 9a

a = 1

Therefore, the required quadratic function is given by:

y = a(x - h)² + k

y = (x - 2)² - 9

Read more on vertex here: brainly.com/question/14946018

#SPJ1

The value of z in the equation 1/4z= -7/8 + 1/8z is -7.

The value of the unknown z in the equation is determined by solving for the unknown from the equation.

The given equation is as follows:

1/4z= -7/8 + 1/8z

To find the value of z in the equation, we simplify the equation.

Multiply both sides by 8 to eliminate the denominators:

8 * (1/4)z = 8 * (-7/8) + 8 * (1/8)z

2z = -7 + z

Next, subtract z from both sides to isolate z:

2z - z = -7

z = -7

Learn more about equations at: https://brainly.com/question/2972832

#SPJ1

The answer is B) 3.250. It is important to note that the critical value depends on the level of significance (c) and the degrees of freedom (n-1).

To find the critical value, tc, for c = 0.99 and n = 10, we need to use the t-distribution table. Since we are dealing with a two-tailed test, we need to find the value that splits the distribution into two parts, each with an area of 0.005 (0.99/2 = 0.495, and 1 - 0.495 = 0.005). Looking at the table, we can see that for 9 degrees of freedom (n-1) and a probability of 0.005, the critical value is 3.250. Therefore, the answer is B) 3.250. It is important to note that the critical value depends on the level of significance (c) and the degrees of freedom (n-1). As the level of significance increases, the critical value increases as well. Similarly, as the degrees of freedom increase, the critical value decreases.

Learn more about critical value here:

https://brainly.com/question/30168469

#SPJ11

The problem involves finding the expected value and variance for the Student t and chi-squared distributions, as well as finding probabilities for certain values of the distributions.

Additionally, the problem requires finding the value of an F distribution with specific degrees of freedom. The expected value for the Student t distribution with v degrees of freedom is 0, and the variance is v/(v-2) when v>2. For the given case with v=21, E(t)=0 and V(t)=21/19=1.1053. The probability of t being greater than 0.859 with 21 degrees of freedom and a significance level of 0.01 is given by t0.01,21 = P(t > 0.859) = 0.1989. The expected value for the chi-squared distribution with v degrees of freedom is v, and the variance is 2v. For the given case with v=9, E(χ2)=9 and V(χ2)=18. The probability of χ2 being greater than 8.343 with 9 degrees of freedom and a significance level of 0.10 is given by χ20.10,9 = 3.325 and P(χ2 > 8.343) = 0.117. The expected value for the F distribution with v1 numerator degrees of freedom and v2 denominator degrees of freedom is v2/(v2-2) when v2>2, and the variance is (2v2^2(v1+v2-2))/((v1(v2-2))^2(v2-4)) when v2>4. For the given case with v1=21 and v2=25, E(F)=1.25 and V(F)=1.9024. The probability of F being less than 0.01 with 21 numerator degrees of freedom and 25 denominator degrees of freedom is F0.01,21,25 = 0.469. To find the value of F0.99,25,21 without using the Distributions tool, we can use the fact that F is the ratio of two independent chi-squared distributions divided by their degrees of freedom, and we can use the inverse chi-squared distribution to find the value. Therefore, F0.99,25,21 = (1/χ2(0.01,21))/(1/χ2(0.99,25)) = 1.5014/0.6793 = 2.211, which is not one of the answer choices provided.

Learn more about chi-squared distributions here: brainly.com/question/31956155

#SPJ11

Let's denote the radius of both the hemisphere and the cylinder as "r" and the height of the cylinder as "h".

The volume of the solid is given as 792π cm^3.

The volume of the hemisphere is (2/3)πr^3, and the volume of the cylinder is πr^2h. Since the solid is made up of a hemisphere and a cylinder, we can write the equation:

(2/3)πr^3 + πr^2h = 792π

We can simplify the equation by dividing both sides by π:

(2/3)r^3 + r^2h = 792

Given that the ratio of the radius to the height is 1:3, we can write r = 3h.

Substituting this value into the equation, we get:

(2/3)(3h)^3 + (3h)^2h = 792

Simplifying further:

(2/3)(27h^3) + 9h^3 = 792

18h^3 + 9h^3 = 792

27h^3 = 792

Dividing both sides by 27:

h^3 = 792/27

h^3 = 29.333...

Taking the cube root of both sides:

h ≈ 3.080

Therefore, the height of the solid is approximately 3.080 cm.

Por lo tanto, la tasa de disminución de la temperatura en grados Celsius por día debe ser menor o igual a 10/3 para que la piscina esté abierta el jueves.

Para resolver este problema, necesitamos utilizar la siguiente fórmula:

Temperatura final = Temperatura inicial - tasa de disminución x días

Sea x la tasa de disminución en grados Celsius por día. Entonces, la temperatura alta el jueves fue:

30 - x(3)

Como se menciona en el problema, la piscina está abierta cuando la temperatura alta es mayor de 20 ∘ C20, degrees, start text, C, end text. Por lo tanto, la temperatura alta el jueves debe ser mayor o igual que 20 ∘ C20, degrees, start text, C, end text. Entonces, tenemos la siguiente desigualdad:

30 - x(3) ≥ 20

Resolviendo para x, obtenemos:

x ≤ 10/3

Por lo tanto, la tasa de disminución de la temperatura en grados Celsius por día debe ser menor o igual a 10/3 para que la piscina esté abierta el jueves.

Learn more about Temperatura here:

https://brainly.com/question/23524228

#SPJ11

The composition of relations a and b on s×s is a∘b={(1,2),(2,2),(2,3),(3,2)}.

To compute the composition of relations a and b, we need to perform the following steps.

First, we need to write out the ordered pairs that are in both a and b. In this case, the only ordered pair that is in both a and b is (1,3).

Next, we need to find all ordered pairs of the form (x,z) such that there exists a y in s such that (x,y) is in b and (y,z) is in a.

In this case, the only such ordered pair is (1,2), since (1,3) is already accounted for.

Finally, we combine the two sets of ordered pairs to get the composition a∘b={(1,2),(2,2),(2,3),(3,2)}.

To learn more about ordered pairs : brainly.com/question/28874341

#SPJ11

The value of x is 8.

Angles that have a total angle of less than 90 degrees are said to be complementary. To put it another way, if two angles combine to form a right angle, that combination is said to be complementary. In this instance, we say that the two angles complement one another well.

In this given figure, we need to find what x is to add up to 90 degrees.

This means that [tex]\sf x^\circ + 82^\circ = 90^\circ[/tex]

[tex]\sf x^\circ=90^\circ-82^\circ[/tex]

[tex]\sf x^\circ=8^\circ[/tex]

Hence, The value of x is 8.

Learn more about complementary angles from the given link

brainly.com/question/98924

We can say that the probability of observing a sample mean of 123.5 wpm or higher by chance alone, assuming the population means is 119 wpm and the standard deviation is 15 wpm, is 4.18%.

To solve this problem, we need to use the central limit theorem, which states that the distribution of sample means will be approximately normal, regardless of the underlying distribution, as long as the sample size is sufficiently large.

In this case, we have a population mean of 119 wpm and a standard deviation of 15 wpm. We want to know the probability that the sample mean would be greater than 123.5 wpm if 33-speed typists are randomly selected.

We can start by calculating the standard error of the mean, which is the standard deviation of the sample mean distribution. We can use the formula:

[tex]$SE = \frac{\sigma}{\sqrt{n}}$[/tex]

where SE is the standard error of the mean, σ is the population standard deviation, and n is the sample size.

Plugging in the values we have:

[tex]$SE = \frac{15}{\sqrt{33}} \approx 2.60$[/tex]

Next, we can calculate the z-score for a sample mean of 123.5 wpm using the formula:

[tex]$z = \frac{\bar{x} - \mu}{SE}$[/tex]

Plugging in the values we have:

z = (123.5 - 119) / 2.60 ≈ 1.73

Using a standard normal distribution table, we can find the probability that the z-score is greater than 1.73. This probability is approximately 0.0418.

Therefore, the probability that the sample mean would be greater than 123.5 wpm if 33-speed typists are randomly selected is approximately 0.0418 or 4.18% (rounded to four decimal places).

To learn more about probability

https://brainly.com/question/32004014

#SPJ4

A possible dimension that will work for the settlement is a length of  186.61 units and a width of  13.39 units.

Area of a rectangle   = L × W

perimeter = 2L + 2W.

we set up equations:

Equation 1: A = L × W = 2,500

Equation 2: P = 2L + 2W < 400

We will solve  this system of equations and find the dimensions

We will arrive at a  quadratic formula:

W = (-b ± √(b² - 4ac)) / (2a)

W = (-(-200) ± √((-200)² - 4(1)(2500))) / (2(1))

W = (200 ± √(40000 - 10000)) / 2

W = (200 ± √30000) / 2

W = (200 ± 173.21) / 2

W₁ = (200 + 173.21) / 2 = 186.61

W₂ = (200 - 173.21) / 2 =13.39

We finally substitute value of w into equation 1

L = 2500 / W

L = 2500 / 13.39 =  186.61

Learn more about quadratic formula at:

https://brainly.com/question/1214333

#SPJ1

A basis for W⊥ is {(2, 6, 1, 0), (0, -2, 0, 1)}. To find a basis for the orthogonal complement W⊥ of the subspace W spanned by w1 and w2, we need to find all vectors that are orthogonal to both w1 and w2.

Let v = (x, y, z, w) be a vector in W⊥. Then we have the following two equations:

w1 · v = 0

w2 · v = 0

where "·" denotes the dot product. Substituting the given vectors and the components of v, we get the following system of linear equations:

x - y + 4z - 2w = 0

y - 3z + w = 0

We can solve this system of equations to find an equation for the plane that contains all vectors orthogonal to W. Adding the two equations, we get:

x - 2z = 0

Solving for x, we get x = 2z. Then substituting into the first equation, we get:

y = 6z - 2w

So a vector v in W⊥ can be written as v = (2z, 6z - 2w, z, w) = z(2, 6, 1, 0) + w(0, -2, 0, 1).

Therefore, a basis for W⊥ is {(2, 6, 1, 0), (0, -2, 0, 1)}.

Visit here to learn more about orthogonal complement brainly.com/question/31669965

#SPJ11

Answer:

Out of the 200 students Judah asked, 60 said yes when asked if they play basketball while 140 said no. To determine the percentage of students who played basketball, we can divide the number of students who said yes by the total number of students and then multiply by 100. 

So, the percentage of students who played basketball is (60/200) x 100 = 30%.

MARK AS BRAINLIEST!!! IT TOOK A LOT OF TIME!

Answer:

The lines are the same. (Infinite Solutions)

Step-by-step explanation:

To solve this, we need to get either x or y to cancel out and equal 0 first.

Let's look at our two equations.

4x-5y=(-2)

-8x+10y=4

I'm going to divide the -8x by 2. Remember when dividing to divide both sides of the equation, otherwise you will end up with something completely different than what you started out with.

-8x+10y=4 (divided by 2)

Our new equation is -4x+5y=2.

Now let's take our other equation. Please see the screenshot to see how this is solved.

Forgive the horrible handwriting, I'm on a computer :(

The container can fit 1600 cube-shaped boxes.

To find out how many cube-shaped boxes can fit into the larger container, we need to calculate the volume of the larger container and divide it by the volume of each cube-shaped box.

The volume of the larger container can be calculated by multiplying its length, width, and height:

Volume of the larger container = 4 ft · 1 1/4 ft · 2 ft

We need to convert the mixed fraction 1 1/4 to an improper fraction:

1 1/4 = (4 · 1 + 1) / 4 = 5/4

Volume of the larger container = 4 ft · (5/4) ft · 2 ft

= (20/4) ft · (5/4) ft · 2 ft

= 50/4 · 2 ft

= 100/4

= 25 ft³

Now let's calculate the volume of each cube-shaped box.

Since all edges are equal to 1/4 ft, the volume can be calculated as the cube of the edge length:

Volume of each cube-shaped box = (1/4 ft)³

= 1/4 ft · 1/4 ft · 1/4 ft

= 1/64 ft³

Finally, we can divide the volume of the larger container by the volume of each cube-shaped box to find out how many boxes can fit:

Number of cube-shaped boxes = (Volume of the larger container) / (Volume of each cube-shaped box)

= 25 / 1/64

= 25 × 64/1

= 1600

Therefore, the container can fit 1600 cube-shaped boxes.

Learn more about volumes click;

https://brainly.com/question/28058531

#SPJ1

The correct answer is c. marginal; average total; minimum. When plotting marginal and average cost curves, the marginal cost curve intersects the average total cost curve at its minimum point.

This is because the average total cost curve is U-shaped, with a downward-sloping portion at low levels of output and an upward-sloping portion at high levels of output. The marginal cost curve intersects the average total cost curve at the point where the upward-sloping portion of the average total cost curve starts, which is also the point where the average total cost curve is at its minimum. At this point, the marginal cost is equal to the average total cost.

Learn more about marginal cost at: brainly.com/question/7781429

#SPJ11

The value of 6⅔ ÷ 12 is 5/9

The mathematical “operation” refers to calculating a value using operands and a math operator.

Mathematical operations include; Addition subtraction, multiplication, division. This operations are used to define the relationship between two terms.

PEDMAS is used for orderly arrangements for the operation.

6²/3 ÷ 12

We first convert the mixed fraction into improper fraction.

= 20/3 × 1/12

= 20/36

divide through by 4, then we have

= 5/9

Therefore the value of 6⅔ ÷ 12 is 5/9

learn more about mathematical operations from

https://brainly.com/question/20628271

#SPJ1

By Using Identity:-

[tex] \quad \hookrightarrow \: { \underline{ \overline{ \boxed{ \pmb{ \sf{ {(a + b)}^{2} = \: {a}^{2} + {b}^{2} + 2ab \: }}}}}} \: \red \bigstar \\ [/tex]

[tex] \sf \longrightarrow \: {(x + 4)}^{2} [/tex]

[tex] \sf \longrightarrow \: {x}^{2} + {4}^{2} + 2 \times 4 \times x[/tex]

[tex] \sf \longrightarrow \: {x}^{2} + {4}^{2} + 8 \times x[/tex]

[tex] \sf \longrightarrow \: {x}^{2} + {4}^{2} + 8 x[/tex]

[tex] \sf \longrightarrow \: {x}^{2} + 16 + 8 x[/tex]

[tex] \sf \longrightarrow \: {x}^{2} + 8 x + 16[/tex]

Therefore ,

________________________________________

[tex] \sf \longrightarrow \: ( x+4 ) ( x-4 )[/tex]

[tex] \sf \longrightarrow \: x ( x - 4 ) + 4( x-4 )[/tex]

[tex] \sf \longrightarrow \: {x}^{2} - 4x + 4x - 16[/tex]

[tex] \sf \longrightarrow \: {x}^{2} - 0 - 16[/tex]

[tex] \sf \longrightarrow \: {x}^{2} -16[/tex]

Therefore,

________________________________________

XY is less than 14,  MR is less than 10.5, if DE || AB, AC = 15, DC = 10, and EC = 8 then BE is equal to 18.75, the sides are not proportional and DE is not parallel to AB.

To find XY, we can use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Using this theorem, we have:

WZ + ZX > WY

6 + 8 > 9

14 > 9

XY must be less than the sum of WZ and ZX. Therefore, XY is less than 14.

To find MR,

RS + ST > RT

6 + 4 1/2 > 3

10.5 > 3

Since the inequality holds true, we can conclude that MR must be less than the sum of RS and ST. Therefore, MR is less than 10.5.

By the similar triangles property:

EC/DC = AC/BC

Substituting the given values:

8/10 = 15/BC

Cross-multiplying:

8 × BC = 10 × 15

BC = 150/8

BC = 18.75

BC=BE

BE is equal to 18.75.

If DE is parallel to AB, then the ratio of the lengths of the corresponding sides AD and BE should be equal.

Using the given lengths:

AD/BE = 4/3

Ratio does not equal 1, which means the sides are not proportional and DE is not parallel to AB.

To learn more on Triangles click:

https://brainly.com/question/2773823

#SPJ1

If a student is ranked in the 85th percentile on their college entrance exams, it means that they scored better than 85% of the other students who took the same exam.

In other words, only 15% of the students who took the exam scored higher than this student. This is a good achievement and suggests that the student is likely to be competitive in the college application process.

A percentile is a statistical metric that shows the proportion of a dataset's values that are equal to or lower than a given value. For instance, the dataset's 75th percentile implies that 75% of the values are equal to or lower than that number.

In the case of the request for the "percentile 100 words," it appears to be a misunderstanding or an incomplete query. In order to produce a useful response, the term "percentile" often needs more details, such as the dataset or the particular value of interest. Could you please elaborate on your point or make your inquiry more clear?

Learn more about percentile here:

https://brainly.com/question/30565593

#SPJ11

Answer:

a) <VZY = (180°- 2×49°)/2 = 41°

b) <VXW = 104°- 41° = 63°

c) <XWZ = 360°- (98°+2×104°) = 54°

Answer:

first option, -1/12

Step-by-step explanation:

slope = rise/run

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

= (-7/2 - -4) / (-2 - 4)

= .5 / -6

=-1/12

Pick the first option

The length of the hypotenuse is 30 units.

The missing length is m = 5.

7.

As per the shown figure, we have

Perpendicular = 15√3

Angle = 60°

We have to determine the length of the hypotenuse.

Using the sine ratio, we can write:

sinθ = Perpendicular / Hypotenuse

Substitute the values in the above formula:

sin60° = 15√3 / Hypotenuse

Hypotenuse = 15√3 / sin60°

Hypotenuse =  15√3 /√3/2

Hypotenuse = 30 units

8.

Using the sine ratio, we can write:

sinθ = P / H

Substitute the values in the above formula:

sin45° = m / 5√2

1/√2 = m / 5√2

m = 5√2/√2

m = 5

Learn more about the trigonometric ratio here:

https://brainly.com/question/31662214

#SPJ1

Function: A mathematical relationship that assigns each input value to a unique output value.

Combined Function: A function formed by applying one function to the output of another function.

Quadratic Function: A function with a polynomial equation of degree 2, represented as f(x) = ax² + bx + c, where a, b, and c are constants.

We have,

Function:

A function is a mathematical relationship or rule that assigns each input value (or element) from a set, called the domain, to a unique output value (or element) from another set, called the range.

Example:

Let's consider a function f(x) = 2x + 3.

This function takes an input value (x), multiplies it by 2, and then adds 3 to get the output value.

For example, if we input x = 4 into the function, we get f(4) = 2(4) + 3 = 11. So, the function maps the input value 4 to the output value 11.

Combined Function:

A combined function is formed by performing multiple operations on a given input value. It involves applying one function to the output of another function.

This allows us to express complex relationships between variables by combining simpler functions.

Example:

Let's consider two functions: f(x) = 2x and g(x) = x².

The combined function h(x) is formed by applying g(x) to the output of f(x). In other words, h(x) = g(f(x)).

If we input x = 3 into the combined function, we first evaluate f(x) = 2(3) = 6, and then evaluate g(6) = 6² = 36. So, h(3) = 36.

Quadratic Function:

A quadratic function is a type of function that can be represented by a polynomial equation of degree 2.

It has the general form f(x) = ax² + bx + c, where a, b, and c are constants.

The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the value of the coefficient "a".

Example:

Let's consider the quadratic function f(x) = 2x² - 3x + 1.

This function has a coefficient of 2 for the x^2 term, -3 for the x term, and 1 for the constant term.

If we input x = 2 into the function,

We get f(2) = 2(2)² - 3(2) + 1 = 8 - 6 + 1 = 3.

So, the function maps the input value 2 to the output value 3.

The graph of this quadratic function is a parabola that opens upwards.

Thus,

Function: A mathematical relationship that assigns each input value to a unique output value.

Combined Function: A function formed by applying one function to the output of another function.

Quadratic Function: A function with a polynomial equation of degree 2, represented as f(x) = ax² + bx + c, where a, b, and c are constants.

Learn more about functions here:

https://brainly.com/question/28533782

#SPJ1

The probability of producing at least 232,000 barrels is 0.5.

The standard normal distribution table, also known as the z-table, is a table that provides the probabilities for a standard normal distribution, which has a mean of 0 and a standard deviation of 1.

The table lists the probabilities for values of the standard normal distribution between -3.49 and 3.49, in increments of 0.01.

Here we have

Daily output of marathon's garyville, louisiana, refinery is normally distributed with a mean of 232,000 barrels of crude oil per day with a standard deviation of 7,000 barrels.

Since the daily output of the refinery is normally distributed,

we can use the standard normal distribution to calculate the probability of producing at least 232,000 barrels.

First, we need to standardize the value using the formula:

=> z = (x - μ) /σ

where:

x = value we want to calculate the probability for (232,000 barrels)

μ = mean (232,000 barrels)

σ = standard deviation (7,000 barrels)

=> z = (232000 - 232000) / 7000 = 0

Next, we look up the probability of producing at least 0 standard deviations from the mean in the standard normal distribution table.

This value is 0.5.

Therefore,

The probability of producing at least 232,000 barrels is 0.5.

Learn more about Normal distribution table at

https://brainly.com/question/30404390

#SPJ4

a) The value of k for which the vectors v₁, v₂, and v₃ are linearly dependent is k = -3.

b) The vectors v₁, v₂, and v₃ form a basis for R³ for any value of k ≠ -3.

c) The system Ax=b is uniquely solvable for each b in R³ for any value of k ≠ -3, and the unique solutions depend on the specific values of b.

a) Linear Dependence:

We have the following system of equations:

c₁ + 2c₂ - c₃ = 0 (Equation 1)

2c₁ + 5c₂ - 4c₃ = 0 (Equation 2)

c₁ + 3c₂ + kc₃ = 0 (Equation 3)

To determine the value of k for linear dependence, we need to solve this system of equations. We can perform row reduction on the augmented matrix [A | 0] to find the row-echelon form.

The augmented matrix [A | 0] is:

| 1 2 -1 | 0 |

| 2 5 -4 | 0 |

| 1 3 k | 0 |

Performing row operations, we can transform the matrix to row-echelon form:

R2 = R2 - 2R1, R3 = R3 - R1

| 1 2 -1 | 0 |

| 0 1 -2 | 0 |

| 0 1 k+1 | 0 |

R3 = R3 - R2

| 1 2 -1 | 0 |

| 0 1 -2 | 0 |

| 0 0 k+3 | 0 |

To have infinitely many solutions, the rank of the augmented matrix [A | 0] must be less than the number of variables (3).

For the rank to be less than 3, the determinant of the remaining matrix must be zero:

det(k + 3) = 0

Solv₁ng det(k + 3) = 0, we find that k = -3.

Therefore, for k = -3, the vectors v₁, v₂, and v₃ are linearly dependent.

b) Basis for R³:

From the previous calculations, we found that for k = -3, the vectors are linearly dependent. Therefore, for k ≠ -3, the vectors are linearly independent.

Next, we need to check if the vectors span R^3. Since we have three vectors, they can span R^3 if their rank is 3.

To find the rank, we can perform row reduction on the matrix [v₁ | v₂ | v₃]:

| 1 2 -1 |

| 2 5 -4 |

| 1 3 k |

Performing row operations, we can transform the matrix to row-echelon form:

R2 = R2 - 2R1, R3 = R3 - R1

| 1 2 -1 |

| 0 1 -2 |

| 0 1 k+1 |

R3 = R3 - R2

| 1 2 -1 |

| 0 1 -2 |

| 0 0 k+3 |

The rank of the matrix [v₁ | v₂ | v₃] is 3 for any value of k ≠ -3.

Therefore, for k ≠ -3, the vectors v₁, v₂, and v₃ form a basis for R^3.

c) Uniquely Solvable System:

For the system Ax=b to be uniquely solvable for each b in R^3, the rank of the augmented matrix [A | b] must be equal to the rank of the coefficient matrix A (which is 3 in this case).

To determine the values of k for which the system is uniquely solvable, we need to check if the augmented matrix [A | b] has a unique row-echelon form.

Let's consider the augmented matrix [A | b] and perform row reduction:

| 1 2 -1 | b₁ |

| 2 5 -4 | b₂ |

| 1 3 k | b₃ |

Performing row operations, we can transform the matrix to row-echelon form:

R2 = R2 - 2R1, R3 = R3 - R1

| 1 2 -1 | b₁ |

| 0 1 -2 | b₂ - 2b₁ |

| 0 1 k+1 | b₃ - b₁ |

R3 = R3 - R2

| 1 2 -1 | b₁ |

| 0 1 -2 | b₂ - 2b₁ |

| 0 0 k+3 | b₃ - b₁ - (b₂ - 2b₁) |

To have a unique solution, the rank of the augmented matrix [A | b] must be equal to the rank of A (which is 3).

For the rank to be 3, the determinant of the remaining matrix must be non-zero:

det(k + 3) ≠ 0

Thus, for k ≠ -3, the system Ax=b is uniquely solvable for each b in R^3. The unique solutions can be obtained by back substitution or using inverse matrices, depending on the specific values of b.

To know more about vector here

https://brainly.com/question/29740341

#SPJ4