Find the equation of the linear function represented by the table below in slope-
intercept form.
X
1
2
3
4
y
5
8
11
14

The solution to the system of equations using matrices is x = 45, y = 15, and z = 30.

A scalar value that can be calculated from a matrix's elements is the determinant. When a square matrix is used to transform vectors, the determinant is a measurement of how much the matrix "stretches" or "shrinks" space. In linear algebra, the determinant is employed in a variety of operations, including as the computation of a matrix's inverse, the description of a matrix's eigenvalues and eigenvectors, and the resolution of linear equation systems. In specifically, the existence of a unique solution, the absence of a solution, or an unlimited number of solutions to a system of linear equations can be determined using the determinant of the coefficient matrix.

The given equation are:

2x -y + 3z = 180

-4x + 2y + 3z = 225

3x - 4y = 270

Writing the equations in matrix form we have:

[tex]\begin{bmatrix} 2 & -1 & 3 \\ -4 & 2 & 3 \\ 3 & -4 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 180 \\ 225 \\ 270 \end{bmatrix}[/tex]

Multiplying the inverse of the coefficient matrix we have:

[tex]\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 2 & -1 & 3 \\ -4 & 2 & 3 \\ 3 & -4 & 0 \end{bmatrix}^{-1} \begin{bmatrix} 180 \\ 225 \\ 270 \end{bmatrix}[/tex]

Now,

[tex]\begin{bmatrix} 2/23 & 5/46 & -3/23 \\ 2/23 & 1/23 & 5/23 \\ -3/23 & -5/46 & 2/23 \end{bmatrix}[/tex]

Multiplying this by the vector on the right-hand side gives:

[tex]\begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 2/23 & 5/46 & -3/23 \\ 2/23 & 1/23 & 5/23 \\ -3/23 & -5/46 & 2/23 \end{bmatrix} \begin{bmatrix} 180 \\ 225 \\ 270 \end{bmatrix} = \begin{bmatrix} 45 \\ 15 \\ 30 \end{bmatrix}[/tex]

Hence, the solution to the system of equations is x = 45, y = 15, and z = 30.

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Check the picture below.

The circumference and approximate area of the given​ circle is 69.08 inches & 380.14 square inches.

What is circumference?

Circumference is the distance around the edge of a circular object or a round shape. It is the length of the boundary or perimeter of the circle. The formula is given by C = 2πr, where C is the circumference, r is the radius of the circle, and π is a mathematical constant approximately equal to 3.14.

The circumference of a circle is given by the formula:

C = 2πr

where r is the radius of the circle and π (pi) is a mathematical constant approximately equal to 3.14.

Using this formula and plugging in the given value of radius:

C = 2 x 3.14 x 11

C = 69.08 inches (rounded to two decimal places)

So the circumference of the circle with 11 inches radius is approximately 69.08 inches.

The area of a circle is given by the formula:

A = πr²

Again, using the given value of radius and approximating π to 3.14:

A = 3.14 x 11²

A = 3.14 x 121

A = 380.14 square inches (rounded to two decimal places)

So the approximate area of the circle with 11 inches radius is approximately 380.14 square inches.

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The probability of the given situation through which the given relation is satisfied is 0.78

Probability is a measure of the likelihood or chance of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates that the event is impossible, and 1 indicates that the event is certain to occur.

The probability of an event can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes. For example, if you flip a fair coin, there are two possible outcomes: heads or tails. The probability of getting heads is 1/2, or 0.5, since there is one favorable outcome (heads) out of two possible outcomes (heads or tails).

Probabilities can also be expressed as percentages or fractions. For example, a probability of 0.25 can be expressed as 25%, or as a fraction of 1/4.

Probability theory is a branch of mathematics that deals with the analysis of random events and the quantification of uncertainty. It has applications in a wide range of fields, including statistics, physics, finance, and engineering.

Flexible hour work = 78%

Don't Know = 9%

Rigid hour = 13%

The probability of that 4 people choose flexible hour for work is,

0.78

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Answer:  33.51 cubic units.

Step-by-step explanation: The equation for the volume of a circle is:

V = (4/3)πr³

where r is the sweep of the circle.

Since the distance across of the circle is given as 4, we are able discover the span by partitioning the distance across by 2:

r = d/2 = 4/2 = 2

Presently we are able plug within the esteem of the sweep into the equation for the volume:

V = (4/3)π(2³) = (4/3)π(8) = 32/3π

Answer: 33.51

Step-by-step explanation:

The formula is 4/3 pi r^3. The radius is 2. If you do the equation, you get roughly 33.51.

The ordered pair (0, 2) lies on the graph.

The ordered pair (-1, 0) lies on the graph.

The ordered pair (-1, 0) lies on the graph.

What is function?

In mathematics, a function is a relationship between two sets of elements, called the domain and the range, such that each element in the domain is associated with a unique element in the range.

The exponential function f(x) = -32x+5 can be written in the form of

f(x) = [tex]a(b^{x})+c[/tex], where a, b, and c are constants. Comparing with the given function, we have a = -3, b = 2, and c = 5.

To find the ordered pairs that lie on the graph of the exponential function, we can plug in different values of x and calculate the corresponding values of y using the function. Here are some examples:

When x = 0, we have f(0) = [tex]-3(2^{0})[/tex] + 5 = 2. Therefore, the ordered pair (0, 2) lies on the graph.

When x = 1, we have f(1) = [tex]-3(2^{1})[/tex] + 5 = -1. Therefore, the ordered pair (1, -1) lies on the graph.

When x = -1, we have f(-1) = [tex]-3(2^{-1})[/tex] + 5 = 6/2 - 3 = 0. Therefore, the ordered pair (-1, 0) lies on the graph.

We can continue this process to find more ordered pairs that lie on the graph of the exponential function.

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Answer:

1. 90

2. 468

Step-by-step explanation:

Answer:

Question 1). 45 in

Question 2). 234 m

Step-by-step explanation:

Area = base x height x 1/2

10 x 9 = 90

90 x 1/2 = 45

Question 2)

30 x 15.6 = 468

468 x 1/2 = 234 m

The negation is the fourth option.

6 + 3 ≠ 9 or 6 - 3 ≠ 9

The negation of an equation is an inequality such that we just change the equal sign, by the "≠" sign.

Here we start with the two equations.

6 + 3 = 9 or 6 - 3 = 9

Just change the equal signs for different signs:

6 + 3 ≠ 9 or 6 - 3 ≠ 9

That is the negation, fourth option.

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The number of red tiles in the box given the chance ratio of red to blue tiles is 18. The surface area of the rectangular prism is 2600 square inches.

Number of red tiles = x

Number of blue tiles = 12 + x

Total tiles = x + 12 + x

= 12 + 2x

Ratio of red = 3

Ratio of blue = 5

Total ratio = 3 + 5 = 8

Number of red tiles = 3 / 8 × 12+2x

x = 3(12 + 2x) / 8

x = (36 + 6x) / 8

8x = 36 + 6x

8x - 6x = 36

2x = 36

x = 36/2

x = 18 tiles

Therefore, The number of red tiles in the box given the chance ratio of red to blue tiles is 18.

b) To find the surface area of the rectangular prism, we need to find the area of each of its faces and add them together. Looking at the net, we see that there are three pairs of identical rectangles: the top and bottom faces, the front and back faces, and the left and right faces. Each of these rectangles has dimensions of 23 inches by 8 inches.

Therefore, the surface area of the rectangular prism is:

=2 * (23 in. * 8 in.) (top and bottom faces)

=2 * (36 in. * 8 in.) (front and back faces)

=2 * (23 in. * 36 in.) (left and right faces)

= 368 + 576 + 1656

= 2600 square inches

Therefore, the surface area of the rectangular prism is 2600 square inches.

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Answer:

$5040

Step-by-step explanation:

Apply the formula

SI = (Principal)(Rate)(Time)

= 6000×0.12×7

= $5,040

Answer:

Try using Special Right Triangles

Step-by-step explanation:

I think it is a carry-on of special right triangles. Since it is 90 degrees let's just assume it is a 60,30,90 triangle. we know the long leg is 6 from the x-6 on the bottom left, and 4rad3 is the hypotenuse.

Long leg: rad 3

Short leg: 1

Hypotenuse: 2

use these ratios to help

I would put 4rad3 across from 2 and y across from 1

aka: 4rad3/y = 2/1

also 4rad3 across from 2 and x across from rad 3

aka:4rad3/x=2/rad3

to find y you simplify

and

to find x also simplify

Answer:

finding the perimeter, you sumthe distance all round that is 7+7.5+17.8+6=38.3

38.3 is the perimeter

Hence,the  sum  of  this  series is  approximately  1.193.

A series in mathematics is   essentially the process of adding an unlimited number of quantities, one after the other, to a specified initial amount. A significant component of calculus and its generalization, mathematical analysis, is the study of series.

If a series partial sum sequence tends to a limit, it is said to be convergent (or to converge); this indicates that if one adds partial sums one after the other in the order indicated by the indices, they approach closer and closer to a specified number.

Comparing  the two series  will help us  better  understand how  they differ.

[tex]\lim_{n \to \infty} \frac{\frac{5}{n(n+3)}}{\frac{1}{n^2}}= \lim_{n \to \infty} \frac{5n^2}{n(n+3)}= \lim_{n \to \infty} \frac{5n}{n+3}= 5[/tex]

Since both series either converge or diverge together,  the limit is a positive finite number. The supplied series [tex]\sum_{n=1}^{\infty} \frac{1}{n^2}[/tex]also  converges because the  series [tex]\sum_{n=1}^{\infty} \frac{5}{n(n+3)}[/tex] converges (by the p-series test with p = 2).

We can employ the partial fraction decomposition to determine the sum:

[tex]\frac{5}{n(n+3)} = \frac{1}{n} - \frac{1}{n+3}[/tex]

Consequently, we have

[tex]\sum_{n=1}^{\infty} \frac{5}{n(n+3)} \\= \sum_{n=1}^{\infty} \left(\frac{1}{n} - \frac{1}{n+3}\right)\\= \left(1 - \frac{1}{4}\right) + \left(\frac{1}{2} - \frac{1}{5}\right) + \left(\frac{1}{3} - \frac{1}{6}\right) + \dots\\[/tex]

[tex]= \frac{3}{4}+ \frac{3}{10}+ \frac{1}{6} + \dots\\[/tex]

The  harmonic  series  of  corresponding  terms  and  the   sequence [tex]0, 0, \frac{1}{6}, 0, 0,\frac{1}{30}, 0, 0, \frac{1}{42}, 0, 0,\frac{1}{66},...[/tex] can  be  added  to  find  the  sum  of  this  series. The  total  is  approximately  1.193.

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Answer: t=2.59

Step-by-step explanation:

This is a matter of clearing out the equation

set 1151=800e^0.14t

1151/800=e^0.14t

ln(1151/800)/0.14=t

t=2.59

The solution of the differential equation y'' + y = t with the initial conditions y(0)=1 and y'(0)=-2 is y(t)= 1-2t+te-t.

Equation is a mathematical statement that expresses the equality of two expressions. It shows the relationship between two or more variables and can be written using symbols, numbers, and operations. Equations are used to describe physical laws, to make calculations, and to solve problems. Examples of equations include the Pythagorean theorem, Newton's laws of motion, and linear equations.

We solve this differential equation using Laplace inverse, with the initial conditions y(0)=1 and y'(0)=-2. First, we take the Laplace transform of the equation:

L[y''+y]=L[t]

Using the properties of Laplace transform, we can write this as:

s2Y(s)-sy(0)-y'(0)+Y(s)= (1/s)

Substituting the initial conditions and rearranging terms, we have:

Y(s)= (1/s) + (2/s2) + (1/s2)

We can then invert the Laplace transform to get the solution of the original equation:

y(t)= 1-2t+te-t

Therefore, the solution of the differential equation y'' + y = t with the initial conditions y(0)=1 and y'(0)=-2 is y(t)= 1-2t+te-t.

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Answer: 5 were triangles :)

The area of given trapezoid whose parallel sides are 8 inches, 16 inches & its perpendicular height is 6 inches is 72 inches².

What is a trapezoid?

Quadrilaterals or four-sided polygons with one pair of parallel sides and one pair of non-parallel sides are known as trapezoids, usually spelt trapezium. It has a perimeter and covers a certain amount of space. The bases of the trapezium are the sides that are parallel to one another. Legs or lateral sides indicates to the non-parallel sides. The altitude or perpendicular height is the separation between the parallel sides. This shape's area is equal to half of the product of its parallel sides times its height.

Given dimensions of trapezoid:

let parallel sides be denoted by a , b & height be 'h'

a= 8 inches (top side)

b= 8+4+4=16 inches (bottom side)

h= 6 inches

Area = [tex]\frac{sum of parallel sides}{2}[/tex] x height

         =[tex]\frac{(a+b)h}{2}[/tex]

         =[tex]\frac{(8+16)6}{2}[/tex]

          =[tex]\frac{24(6)}{2}[/tex]

          =24 x 3

          =72 sq. inches

The area of given trapezoid is 72 inches²

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The polynomial function of least degree that has zeros of x=0, x=2i, and x=3, and with all coefficients real is:

Since the zeros of the polynomial function are given as

x=0, x=2i, and x=3,

we can write the function in factored form as follows:

f(x) = a(x-0)(x-2i)(x-3)

where

a is a constant coefficient and the factors correspond to the given zeros.

Since all coefficients must be real, we know that the complex conjugate of 2i, which is -2i, must also be a zero of the function. Therefore, we can rewrite the function as:

f(x) = a(x-0)(x-2i)(x+2i)(x-3)

Expanding this expression gives:

f(x) = a(x² + 4)(x-3)

Multiplying out the brackets and collecting like terms, we get:

f(x) = ax³ - 3ax² + 4ax - 12a

To find the value of 'a', we can use the fact that the coefficient of the x³ term is 1. Thus, we have:

a = 1/(1*4) = 1/4

Substituting this value of 'a' in the above expression, we get:

f(x) = (1/4)x³ - (3/4)x² + x - 3

Therefore, the polynomial function of least degree that has zeros of x=0, x=2i, and x=3, and with all coefficients real is:

Option A: f(x) = x² - 3x³ + 4x² - 12x

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Answer:

$60

Step-by-step explanation:

In a set of data, the median is the value that is placed in the middle. to find the median, we must order the numbers from least to greatest, like so:

20, 40, 40, 50, 60, 60, 60, 60, 70, 70

Using this representation, we find that there are two numbers in the middle: 60 and 60 (since the amount of data is an even number, there will be two numbers in the center).

When something like this occurs, find the average of the two numbers in the middle (and the values and divide by the number of values). We have:

60 + 60 = 120;

[tex]\frac{120}{2} = 60[/tex]

Therefore, the median is $60.

Answer:

p=42

Step-by-step explanation:

S=104 sm^2

a=8 sm

S=a*b

104 sm^2=8*b

b=104 sm^2/8 sm=13 sm

P=2*(a+b)=2*(13 sm+8 sm)=42 sm

The outcome that Regis is likely to get after randomly drawing one tile from the bag would be 0. That is option A.

To calculate the outcome of the event is to calculate the probability of selecting a tile with a number when one tile is drawn at random.

Probability = possible outcome/sample space.

Possible outcome = 1

sample space = 8

probability = 1/8 = 0.125

The probability is approximately = 0

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The intervals on which the function Is constant are [-4, -3] and [3, 6]

A function is considered constant over an interval if the function has the same output for all the inputs within that interval.

In other words, the function does not change over that interval.

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

The function does not change over the intervals [-4, -3] and [3, 6]

Hence, the intervals are [-4, -3] and [3, 6]

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The value of p(B/A) represents the probability that event B will occur given that event A has occurred.

It is a measure of conditional probability, which is calculated using the formula:

p(B/A) = p(A ∩ B) / p(A)

Where:

Therefore, To calculate the value of p(B/A), you need to know the probabilities of A and B, as well as the probability that both occur together.

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The correct answer is (1,1) because both of the lines meet together at these numbers

The equation of the plane that passes through the line of intersection of x - z = 3 and y + 3z = 3 and is perpendicular to x + y - 4z = 6 is x + y - 4z = 3.

The point-normal form of the equation of a plane is given by:

N · (<x - x0>, <y - y0>, <z - z0>) = 0

Where (x0, y0, z0) is a point on the plane and N = is a normal vector to the plane, we have the point-normal form of the equation of a plane. The dot product of the vector from the supplied location to any point on the plane with the normal vector to the plane yields this form of the equation. The equation states that any vector located in the plane with the normal vector has a zero dot product. The scalar equation of the plane can also be found by expanding the dot product, and it takes the form axe + by + cz = d, where d = N (x0, y0, z0).

Given the equation of the planes is x − z = 3 and y + 3z = 3.

Now, find the direction vector of the line of intersection:

Set z = t:

x = t + 3 and y = 3 - 3t

The direction vector is <1, -3, 1>.

2. Determine the normal vector:

The plane is perpendicular to the plane x + y - 4z = 6, so:

normal vector of x + y - 4z = 6, which is <1, 1, -4>.

3. Using point normal form we have:

(3, 0, 0)

The point satisfies the equation:

x - z = 3 and y + 3z = 3 when z = 0

Thus,

<1, 1, -4> · <x - 3, y, z> = 0

x + y - 4z = 3

Hence, the equation of the plane that passes through the line of intersection of x - z = 3 and y + 3z = 3 and is perpendicular to x + y - 4z = 6 is x + y - 4z = 3.

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The equation of the normal plane is 4y = 4π, or equivalently, y = π.

The word osculate comes from the Latin osculatus, which is a past participle of the verb osculari, which means "to kiss." Thus, an osculating plane is one that "kisses" a submanifold.

To find the osculating plane and normal plane of the curve x = sin 2t, y = t, z = cos 2t at the point (0, π, 1), we need to follow these steps:

Step 1: Find the first and second derivatives of the curve with respect to t.

x' = 2cos2t

y' = 1

z' = -2sin2t

x'' = -4sin2t

y'' = 0

z'' = -4cos2t

Step 2: Evaluate the derivatives at t = π.

x'(π) = 2cos2π = 2

y'(π) = 1

z'(π) = -2sin2π = 0

x''(π) = -4sin2π = 0

y''(π) = 0

z''(π) = -4cos2π = -4

So the velocity vector at the point (0, π, 1) is v = ⟨2, 1, 0⟩, the acceleration vector is a = ⟨0, 0, -4⟩, and the curvature vector is κv = ⟨0, 4, 0⟩.

Step 3: Use the velocity and acceleration vectors to find the normal vector of the osculating plane.

The normal vector of the osculating plane is given by the cross product of the velocity and acceleration vectors:

n = v × a = ⟨2, 1, 0⟩ × ⟨0, 0, -4⟩ = ⟨4, 0, 0⟩

Step 4: Use the normal vector and the point (0, π, 1) to find the equation of the osculating plane.

The equation of the osculating plane is given by:

4(x - 0) + 0(y - π) + 0(z - 1) = 0

Simplifying, we get:

4x - 4 = 0

So the equation of the osculating plane is 4x = 4, or equivalently, x = 1.

Step 5: Use the curvature vector to find the normal vector of the normal plane.

The normal vector of the normal plane is given by the curvature vector:

n' = κv = ⟨0, 4, 0⟩

Step 6: Use the normal vector and the point (0, π, 1) to find the equation of the normal plane.

The equation of the normal plane is given by:

0(x - 0) + 4(y - π) + 0(z - 1) = 0

Simplifying, we get:

4y - 4π = 0

So, the equation of the normal plane is 4y = 4π, or equivalently, y = π.

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Answer:

Axis of symmetry =1

vertex =(1,2)

y intercept =0

min/max= -6,2

domain= 0,1,2

range =y≥1,2

According to the given information, the function has no cusp.

What is a function?

A function is a relation between a set of inputs and a set of possible outputs, with the property that each input is related to exactly one output.

The given piecewise function is:

c(x) = 39, when x ≤ 6

c(x) = 39 + 5(x - 6), when x > 6

A cusp is a point on the graph where the function changes direction very abruptly, like a sharp turn. This happens when the derivative of the function is not defined at that point.

The derivative of the function is:

c'(x) = 0, when x ≤ 6

c'(x) = 5, when x > 6

Since the derivative is defined and continuous at x = 6, there is no cusp at that point. Therefore, the function has no cusp.

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Answer:

A parallelogram with congruent diagonals, a parallelogram with a right angle, a parallelogram with perpendicular diagonals.

Step-by-step explanation:

By definition, a rectangle is nothing but a parallelogram with one right angle. One of the properties of a rectangle is that its perpendiculars are congruent. Finally, the diagonals of a rectangle are congruent (this can be proved by finding congruent triangles split by the diagonals, using CPCTC, etc.).