2.) 0.928
As scientific notation

Answer:

Make a line graph with all the data. I think

Step-by-step explanation:

9.5 To find matrix P and diagonal matrix D such that [tex]P^{-1}AP = D[/tex], the given matrix A needs to be diagonalized.

9.5 [tex]A^{23}[/tex] can be calculated by multiplying A to itself 23 times.

9.6 The system of differential equations can be solved using the diagonalized form of A obtained in question 9.4.

9.4 To find matrix P and diagonal matrix D, we need to diagonalize the given matrix A. Diagonalization involves finding the eigenvalues and eigenvectors of A. By calculating the eigenvalues and eigenvectors, we can construct matrix P using the eigenvectors as its columns. The diagonal matrix D is formed by placing the eigenvalues on the diagonal. The obtained matrices P and D satisfy the equation [tex]P^{-1}AP = D[/tex].

9.5 To find [tex]A^{23}[/tex], we can multiply the matrix A to itself 23 times. The resulting matrix will have the values of A raised to the power of 23.

9.6 The system of differential equations can be solved using the diagonalized form of matrix A obtained in question 9.4. Since A = [tex]PDP^{-1}[/tex], we can rewrite the system of differential equations in terms of P and D. Then, by performing a change of variables using [tex]P^{-1}[/tex], the system can be transformed into a diagonal system. This diagonal system can be solved by solving each diagonal equation separately. The solutions can then be transformed back to the original variables using P.

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The correct equation from the diagram in this problem is given as follows:

x = 17.3/cos(37º).

The three trigonometric ratios are the sine, the cosine and the tangent of an angle, and they are obtained according to the formulas presented as follows:

For the angle of 37º, we have that:

Hence the equation is obtained as follows:

cos(37º) = 17.3/x

x = 17.3/cos(37º).

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Answer:

Option (D)

Step-by-step explanation:

Rate of emptying the fuel tank = 7.5 liters per minute

Fuel emptied in x minute = 7.5x liters

Volume of the fuel in the fuel tank initially = 175 liters

Therefore, amount of fuel left in the tank after x minutes = (175 - 7.5x) liters

Function representing the emptying of the fuel tank will be,

f(x) = 175 - 7.5x

Option (D) will be the answer.

Answer:

gradient = 3

Step-by-step explanation:

Calculate the gradient ( slope) using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = (2, 3) and (x₂, y₂ ) = (4, 9)

m = [tex]\frac{9-3}{4-2}[/tex] = [tex]\frac{6}{2}[/tex] = 3

The bearing of D from C is equal to the angle measure 315°.

Bearing is usually measured in degrees, with 0° indicating the reference direction (usually North), and increasing clockwise to 360°. It refers to the direction or angle between a reference direction and a point or object.

By observation, the angle between the north line CN and the line CD is equal to 45°, the bearing of D from C is the angle formed beginning from the north line clockwise to the line CD calculated as:

360° - 45° = 315°

Therefore, the bearing of D from C is equal to angle measure 315°

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Answer:

The slope is 1.

Step-by-step explanation:

Mr Morake's Municipal bill showed R201,27 for water Usage at the end of August 2018 and he stated that the basic charge was not included on the water bill, then he is incorrect.

Mr Morake's municipal bill showed R201,27 for water usage at the end of August 2018. He stated that the basic charge was not included on the water bill. Verify if this statement is correct.

A basic charge is a fixed fee that appears on a water bill regardless of the volume of water used. It covers the cost of the water distribution system, maintenance of the infrastructure, and customer support. It's not linked to your actual water usage and is included in every monthly bill, according to most municipal billing systems.

What is a basic charge?

A basic charge is a fixed fee that is charged for water usage. The fee is fixed and does not change from month to month. It is a fee that is typically charged by municipalities. It is used to help cover the costs of maintaining the water infrastructure.

The basic charge includes the following costs:

Water infrastructure management Pumping stations Reservoirs Pipeline maintenance meter maintenance Customer service This charge ensures that the water system is maintained and that customers receive excellent service, even if their water usage is low.

So, if Mr Morake's municipal bill showed R201,27 for water usage at the end of August 2018 and he stated that the basic charge was not included on the water bill, then he is incorrect. As previously stated, the basic charge is included on every monthly bill, according to most municipal billing systems.

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The angle ZB can be computed using the dot product of vectors ZB and ZC, and the area of the triangle can be found using the cross product of vectors ZB and ZC.

To find the angle ZB, we first calculate the vectors ZB and ZC:

[tex]\(ZB = B - Z = (-1, 4, 1) - (4, 10, 7) = (-5, -6, -6)\)[/tex]

[tex]\(ZC = C - Z = (3, 7, 2) - (4, 10, 7) = (-1, -3, -5)\)[/tex]

Then, we calculate the dot product of ZB and ZC:

[tex]\(ZB \cdot ZC = (-5, -6, -6) \cdot (-1, -3, -5) = 43\)[/tex]

Using the dot product formula, we can find the angle between the two vectors:

[tex]\(\cos(ZB) = \frac{{ZB \cdot ZC}}{{\|ZB\| \cdot \|ZC\|}} = \frac{{43}}{{\sqrt{77} \cdot \sqrt{35}}}\)[/tex]

Finally, we can calculate the angle ZB using the inverse cosine function:

[tex]\(ZB = \cos^{-1}\left(\frac{{43}}{{\sqrt{77} \cdot \sqrt{35}}}\right)\)[/tex]

To find the area of the triangle formed by the three points, we calculate the cross product of vectors ZB and ZC:

[tex]\(ZB \times ZC = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -5 & -6 & -6 \\ -1 & -3 & -5 \end{vmatrix}\)[/tex]

By expanding the determinant, we obtain the resulting vector:

[tex]\(ZB \times ZC = (12, -1, -9)\)[/tex]

The magnitude of this vector represents the area of the triangle:

[tex]\(\text{Area} = \left\|ZB \times ZC\right\|\)[/tex]

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Troy should include $1,200 in gross income related to his gambling activities.

Nicci claimed the standard deduction, she can exclude the $720 refund from her gross income.

For Troy, the amount he should include in gross income related to his gambling activities is the net winnings, which is calculated by subtracting his losses from his winnings.

Net Winnings = Winnings - Losses

Net Winnings = $2,500 - $1,300

Net Winnings = $1,200

Therefore, Troy should include $1,200 in gross income related to his gambling activities.

For Nicci, if she claimed the standard deduction on her 2021 tax return, she does not need to include the state refund in her gross income. State tax refunds are generally not taxable if the taxpayer did not itemize deductions in the year for which the refund is received. Since Nicci claimed the standard deduction, she can exclude the $720 refund from her gross income.

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The cross product, denoted by ×, of two vectors x and w in R³ is a vector that satisfies the vector equation x = sv + tw. This can be explained by the geometric properties of the cross-product in three-dimensional space. The cross product produces a vector that is perpendicular (orthogonal) to both x and w, which makes it a natural choice to express x as a linear combination of v and w.

In three-dimensional space, the cross product of two vectors x and w, denoted by x × w, produces a vector that is perpendicular to both x and w. This means that x × w is orthogonal to both x and w.

Now, consider the vector equation x = sv + tw, where s and t are scalars. We want to find a vector that satisfies this equation. Since x × w is orthogonal to both x and w, it is also orthogonal to any linear combination of x and w, including sv and tw. Therefore, x × w is a natural choice as the vector that satisfies the equation x = sv + tw.

Geometrically, x can be seen as the sum of the vector sv, which lies along the same line as v, and the vector tw, which lies along the same line as w. The cross product x × w provides a vector that is perpendicular to both v and w, completing the span of vectors that can be formed by linear combinations of v and w.

In conclusion, the cross product x × w is a vector that satisfies the vector equation x = sv + tw due to its orthogonality to both x and w.

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Answer:

127.75

Step-by-step explanation:

64+32+16+8+4+2+1+.5+.25=127.75

Answer:

127.75

Step-by-step explanation:

Approximately, the value of the series is -7,619,287 to within an error of at most [tex]10^-5.[/tex]

The series is

[tex](-1)^(n+1) * n^6 n=1.[/tex]

We have to approximate the value of the series to within an error of at most 10^-5.

To obtain an approximation for the series, let us evaluate the sum S6.

Hence, the series S is given as:

[tex]S = (-1)^(1+1)*1^6 + (-1)^(2+1)*2^6 + (-1)^(3+1)*3^6 + ... + (-1)^(n+1)*n^6[/tex]

To obtain an approximation to this series, we will use the Alternating Series Estimation Theorem, which is given by:

|SN - S| <= |Rn|, where Rn is the remainder when SN is used to approximate the sum S.

Therefore, we have to determine the sum S6, the alternating series SN6, and the remainder R6.

|S6 - SN6| = R6

= |-7,605,967|

≈[tex]7.61 x 10^6.[/tex]

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Answer:  A

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The size of the angle marked x is 105 degrees. The reason for this answer is that alternate Angles formed between two parallel lines are always equal.

To find the size of the angle marked x, we need to understand the concept of alternate angles and parallel lines. According to the given statement, AB is parallel to CD, which means the lines AB and CD never intersect each other and they remain at the same distance from each other. Also, two lines intersected by a transversal form various pairs of angles such as alternate angles, corresponding angles, interior angles, and exterior angles. Alternate angles are equal when two parallel lines are intersected by a transversal.

Using this property of alternate angles, we can say that angle x is equal to angle z. It is because both angle x and angle z are alternate angles and they are formed between the parallel lines AB and CD. So, we can write:

angle x = angle z Now, we can use the given information to find the value of angle z. We are given that angle z and angle y add up to 180 degrees. It is because angle z and angle y are adjacent angles and they are formed on a straight line. So, we can write:

angle z + angle y = 180 degreesWe know that angle y is equal to 75 degrees. We can substitute this value in the above equation and solve for angle z.angle z + 75 = 180 Subtracting 75 from both sides, we get:angle z = 180 - 75angle z = 105 degrees

Now that we have found the value of angle z, we can use the property of alternate angles to find the value of angle x. We have already established that angle x is equal to angle z. So, we can write:angle x = 105 degrees

Therefore, the size of the angle marked x is 105 degrees. The reason for this answer is that alternate angles formed between two parallel lines are always equal.

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The median speed at which Dave drove to work is 62 mph.

To find the median speed at which Dave drove to work, we need to arrange the average speeds in ascending order and determine the middle value.

Arranging the average speeds in ascending order:

44, 45, 54, 59, 62, 63, 66, 67, 70

Since we have nine values, the middle value is the 5th value, which is 62.

Therefore, Dave's average speed on the way to work was 62 mph.

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The equation has a slope of -2 and the value of 5 is the y-intercept.

When ever you need to find the y-intercept, you need to look at the number without x.

Answer:

see below

Step-by-step explanation:

<2 and <3 are same side interior angles

Since p and q are parallel, <2 and <3 add to 180 ( or are supplementary)

Answer:

They are consecutive interior angles

Step-by-step explanation:

Answer:

x = -2

y = 8

Step-by-step explanation:

8x + 5y = 24

y = -4x

Substitute y;

8x + 5(-4x) = 24

Distribute;

8x - 20x = 24

-12x = 24

Divide both sides by -12

x = -2

Substitute x;

y = -4(-2)

y = 8

The solution to the system of equations is x = -2 and y = 8.

To solve the system of equations:

8x + 5y = 24

y = -4x

We can use the second equation (y = -4x) to substitute for 'y' in the first equation. This process is known as substitution. Replace 'y' in the first equation with (-4x):

8x + 5(-4x) = 24

Now, simplify and solve for 'x':

8x - 20x = 24

Combine the x terms:

-12x = 24

Next, isolate 'x' by dividing both sides by -12:

x = 24 / -12

x = -2

Now that we have the value of 'x', we can find 'y' by substituting '-2' back into the second equation:

y = -4(-2)

y = 8

The solution to the system of equations is x = -2 and y = 8.

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The equation can be interpreted as follows: The salary (y) of the employee is equal to the initial salary (25,000) multiplied by the total percentage increase in the salary (1 + 0.04).

In the equation y = 25,000(1 + 0.04), each number and variable has a specific meaning:

y: This represents the salary of the employee after receiving the annual raise. It is the dependent variable that we are trying to determine.

25,000: This is the initial salary of the employee before the annual raise. It is a constant value and represents the starting point for the salary calculation.

1: This represents the base value of 100% or 100% of the initial salary. It is added to the raise percentage to calculate the total percentage increase in the salary.

0.04: This represents the raise percentage, which is 4% in this case. It signifies that the employee's salary will increase by 4% of the initial salary.

By calculating this equation, we can determine the final salary of the employee after the annual raise.

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The side lengths of the right triangle are: shorter leg = 7 cm, longer leg = 24 cm, and hypotenuse = 25 cm.

Let's denote the length of the shorter leg as x.

According to the information:

The length of the longer leg is 3 cm more than three times the length of the shorter leg, which can be expressed as 3x + 3.

The length of the hypotenuse is 4 cm more than three times the length of the shorter leg, which can be expressed as 3x + 4.

In a right triangle, the Pythagorean theorem states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Using this, we can set up the equation:

(x)²+ (3x + 3)² = (3x + 4)²

Expanding and simplifying the equation:

x² + (9x² + 18x + 9) = (9x² + 24x + 16)

Combining like terms:

10x² + 18x + 9 = 9x² + 24x + 16

Moving all terms to one side of the equation:

10x² + 18x + 9 - 9x² - 24x - 16 = 0

Simplifying:

x² - 6x - 7 = 0

Now, we can solve this quadratic equation by factoring or using the quadratic formula. Factoring, we have:

(x - 7)(x + 1) = 0

Setting each factor to zero:

x - 7 = 0   or   x + 1 = 0

Solving for x:

x = 7   or   x = -1

Since lengths cannot be negative, we discard the solution x = -1.

Therefore, the length of the shorter leg is x = 7 cm.

Using this value, we can find the length of the longer leg and the hypotenuse:

Length of the longer leg = 3x + 3 = 3(7) + 3 = 21 + 3 = 24 cm

Length of the hypotenuse = 3x + 4 = 3(7) + 4 = 21 + 4 = 25 cm

So, the side lengths of the triangle are:

Shorter leg = 7 cm

Longer leg = 24 cm

Hypotenuse = 25 cm

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Answer:

D

Step-by-step explanation:

Answer:

x=5

Step-by-step explanation:

-1+x=4

+1   +1

x=5

hope this helps :)

Matrix A can be diagonalized by using the Orthogonal matrix P and the diagonal matrix D as follows: A = PDP^-1, where P^-1 = PT.

a) The characteristic equation of the matrix A is as follows:(-2-p)(p2−6p−16) = 0, where p is used instead of x.

b) The eigenvalues of the matrix are -2, -2, and 8, including repetition.c)

The process of orthogonal diagonalization is as follows, where P is an orthogonal matrix and D is a diagonal matrix that contains the eigenvalues of A along the diagonal

:First, we need to find the eigenvectors that correspond to each of the eigenvalues of A.i) For λ = -2: A + 2I = 2 20 30 −3 row2-2row1: A + 2I = 2 20 30 0 0 0 -3row2+row3: A + 2I = 2 20 30 0 -3 0 ii) For λ = 8: A - 8I = -8 -20 30 -1 row2 + row1: A - 8I = -8 -20 30 -51 row3 - 3row2: A - 8I = -8 -20 30 -51 iii) For λ = -2: A + 2I = 2 20 30 −3 row2-2row1: A + 2I = 2 20 30 0 0 0 -3row2+row3: A + 2I = 2 20 30 0 -3 0

Therefore, we have the eigenvectors for λ = -2, which are [-3/√10, 1/√10, 0] and [3/√10, 1/√10, 0].We also have the eigenvector for λ = 8, which is [1/√2, 0, -1/√2].

Thus, the orthogonal matrix P is given by: P = [−3/√10 3/√10 1/√2 1/√10 1/√10 0 0 0 -1/√2]

Finally, the diagonal matrix D is: D = [−2 0 0 0 −2 0 0 0 8].

Therefore, matrix A can be diagonalized by using the orthogonal matrix P and the diagonal matrix D as follows: A = PDP^-1, where P^-1 = PT.

The answer in the matrix is given below: $A=\begin{bmatrix}-2 & 0 & 0\\0 & -2 & 0\\0 & 0 & 8\end{bmatrix}.

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Answer:

y=1/2x -10

Step-by-step explanation:

use y2-y1/x2-x1 and solve

than y-y1=answer(x-x1)

Answer:

A. 63 degrees

Step-by-step explanation:

63+63+54=180