Find the numerical value of the log expression.
log a = -1
log b = -6
log
a²
64c4
log c = 3

The solution to the inequality |x − 2| ≥ 5 is x ≤ -3 or x ≥ 7.

We have,

To solve the inequality |x − 2| ≥ 5, we need to consider two cases:

Case 1:

x - 2 is non-negative (i.e., x - 2 ≥ 0)

In this case, the absolute value simplifies to x - 2, so we have:

x - 2 ≥ 5

Adding 2 to both sides, we get:

x ≥ 7

So any value of x that is greater than or equal to 7 satisfies the inequality.

Case 2:

x - 2 is negative (i.e., x - 2 < 0)

In this case, the absolute value simplifies to -(x - 2), so we have:

-(x - 2) ≥ 5

Multiplying both sides by -1, we get:

x - 2 ≤ -5

Adding 2 to both sides, we get:

x ≤ -3

So any value of x that is less than or equal to -3 satisfies the inequality.

Therefore,

The solution to the inequality |x − 2| ≥ 5 is x ≤ -3 or x ≥ 7.

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The other zero of the function given that one zero is 4 include: B. 8.

The correct graph of the quadratic function represented in the table is: graph A.

In Mathematics and Geometry, the standard form of a quadratic function is represented by the following equation;

ax² + bx + c = 0

Next, we would create a system of equation by using the data points provided in the table above;

a(2)² + b(2) + c = 12

4a + 2b + c = 12     .....equation 1.

a(6)² + b(6) + c = -4

36a + 6b + c = -4     .....equation 2.

a(7)² + b(7) + c = -3

49a + 7b + c = -3     .....equation 3.

By solving the system of equations simultaneously, the values of a, b, and c are as follows;

a = 1

b = -12

c = 32

Therefore, the required quadratic function is given by;

ax² + bx + c = 0

x² - 12x + 32 = 0

x² - 8x - 4x + 32 = 0

x(x - 8) - 4(x - 8) = 0

(x - 4)(x - 8) = 0

x = 4 or x = 8

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The reduced row echelon form of the matrix is given as follows:

[tex]\left[\begin{array}{cccc}1&0&0&8\\0&1&0&-2\\0&0&1&-6\end{array}\right][/tex]

The row echelon form of a matrix contains the entire matrix except the last column in the format of an identity matrix, with the last column containing the solution to the system of equations represented by the matrix.

The matrix in the context of this problem is defined as follows:

[tex]\left[\begin{array}{cccc}-1&5&-3&6\\0&3&6&-42\\0&-6&-1&18\end{array}\right][/tex]

First we want a value of 1 at line 1, column 1, hence we multiply the first line by -1, that is:

R1 -> -R1.

Hence:

[tex]\left[\begin{array}{cccc}1&-5&3&-6\\0&3&6&-42\\0&-6&-1&18\end{array}\right][/tex]

Then we want a value of 1 at line 2 column 2, thus:

R2 -> R2/3.

Hence:

[tex]\left[\begin{array}{cccc}1&-5&3&-6\\0&1&2&-14\\0&-6&-1&18\end{array}\right][/tex]

Then we want a value of 0 at line 3, column 2, hence:

L3 -> L3 + 6L2

[tex]\left[\begin{array}{cccc}1&-5&3&-6\\0&1&2&-14\\0&0&11&-66\end{array}\right][/tex]

Then the solution to the system of equations is given as follows:

Hence the row echelon form of the matrix is given as follows:

[tex]\left[\begin{array}{cccc}1&0&0&8\\0&1&0&-2\\0&0&1&-6\end{array}\right][/tex]

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The federal government spends slightly more on defense than Social Security.

We have,

The federal government spends different amounts of money on different programs and services every year, so there is no fixed answer to this question.

However, based on recent data, we can make some general statements about federal spending on defense and Social Security.

In the United States, defense spending typically makes up a significant portion of the federal budget, usually around 15-20%.

Social Security spending is also a major component of the federal budget, making up about 24% of total federal spending in recent years.

Thus,

Based on these figures, we can say that the federal government spends slightly more on Social Security than on defense.

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Answer:

[tex]\frac{8}{52}[/tex]

Step-by-step explanation:

given

[tex]\frac{2}{13}[/tex]

to obtain a denominator of 52 multiply the original denominator of 13 by 4 , that is 13 × 4 = 52

the numerator must also be multiplied by 4, then

[tex]\frac{2}{13}[/tex] = [tex]\frac{2(4)}{13(4)}[/tex] = [tex]\frac{8}{52}[/tex]

The vertices of the triangle XYZ after a dilation with a scale factor of [tex]4[/tex] are [tex]X'(-8, 4)[/tex], [tex]Y'(12, -4)[/tex], and [tex]Z'(4, 16)[/tex].

To find the vertices of the image after a dilation with a scale factor of [tex]4[/tex], we need to multiply the coordinates of each vertex by the scale factor.

Given the triangle with vertices [tex]X(-2,1), Y(3,-1), and \ Z(1,4)[/tex], we can apply the dilation to each vertex.

Let's start with vertex [tex]X(-2,1)[/tex]:

The new x-coordinate,[tex]\(x'\)[/tex], can be obtained by multiplying the original x-coordinate by the scale factor: [tex]\(x' = 4 \times (-2) = -8\)[/tex].

Similarly, the new y-coordinate, [tex]\(y'\)[/tex], is obtained by multiplying the original y-coordinate by the scale factor: [tex]\(y' = 4 \times 1 = 4\)[/tex].

Thus, the image of vertex X is [tex]X'(-8, 4)[/tex].

Now let's apply the same process to vertices [tex]Y(3,-1) \ and \ Z(1,4):[/tex]

For vertex Y:

[tex]\(x' = 4 \times 3 = 12\)\\\y' = 4 \times (-1) = -4\)[/tex]

The image of vertex Y is Y'(12, -4).

For vertex Z:

[tex]\(x' = 4 \times 1 = 4\)\\\(y' = 4 \times 4 = 16\)[/tex]

The image of vertex Z is [tex]Z'(4, 16)[/tex].

Therefore, the vertices of the triangle XYZ after a dilation with a scale factor of [tex]4[/tex] are [tex]X'(-8, 4)[/tex], [tex]Y'(12, -4)[/tex], and [tex]Z'(4, 16)[/tex].

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Answer:

The scale of the drawing is 1 cm to 90 m. This means that every 1 cm on the drawing represents 90 m in real life. The length of the field is 450 m, so the length of the drawing will be 450 m / 90 m = 5 cm.

Therefore, the length of the dam drawing is 5 cm.

Step-by-step explanation:

From the figure, the blanks are filed as follows

Common internal tangent: is matched to line HJ

point of tangency: is matched to G

Center: is matched to A

chord: is matched to line GK

secant: is matched to line CD

diameter: is matched to line EF

common external tangent = line KF

radius: is matched to line BF

semi-circle = arc EGF

Major arc: is matched to arc CDK

Minor arc: is matched to arc CK

In geometry the point of tangency is the point where a line or curve touches a circle or other curved shape at exactly one point without crossing over or intersecting it at another point.

At the point of tangency the tangent line is perpendicular to the radius of the circle that intersects it.

In the figure, the point of tangency is at F E G and K

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part a.

The average velocity on the interval [0,2] is -16 ft/sec

part b.

The instantaneous velocity when t=2  is found as -48 ft/sec

part c.

Time taken to reach the ground is 3 seconds.

part d.

velocity of the ball when it hits the ground is found as  80 ft/sec downward.

The average velocity on the interval [0,2] is determined by :

average velocity = change in distance / change in time

change in distance :

s(2) - s(0) = (-16(2)^2 + 16(2) + 96) - (-16(0)^2 + 16(0) + 96) = -32

where the change in time:

2 - 0 = 2

average velocity = -32/2

average velocity = -16 ft/sec

b. The instantaneous velocity :

s'(t) = -32t + 16

and  t=2:

s'(2) = -32(2) + 16 =

s'(2)=  -48 ft/sec

the velocity of the ball when it hits the ground is at time = 3

s'(t) = -32t + 16

s'(3) = -32(3) + 16 =

s'(3)  =  -80 ft/sec

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Sam started with £5.

Now, we need to find out how much money Sam has left after spending £4.50 on chocolate and £3.00 on cold coffee. To do this, we can add the two amounts that he spent and subtract the total from his original amount of money:

£4.50 + £3.00 = £7.50

Original amount - £7.50 = Two-fifths of the original amount

Now, we know that Sam has two-fifths of his original amount of money left. We can use this information to set up an equation:

2/5 x Original amount = Amount of money left

We can solve for the original amount by multiplying both sides by 5/2:

Original amount = (5/2) x Amount of money left

Plugging in the amount of money Sam has left, we get:

Original amount = (5/2) x Amount of money left

Original amount = (5/2) x (Original amount - £7.50)

Simplifying the right side of the equation, we get:

Original amount = (5/2) x Original amount - (5/2) x £7.50

Original amount - (5/2) x Original amount = (5/2) x £7.50 (3/2) x Original amount = (15/2)

Original amount = £5

Therefore, Sam started with £5.

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The time needed to pay the compound interest of Rs 6,305 is given as follows:

3 years.

The amount of money earned, in compound interest, after t years, is given by:

[tex]A(t) = P\left(1 + \frac{r}{n}\right)^{nt}[/tex]

In which:

The parameters for this problem are given as follows:

P = 40000, A(t) = 40000 + 6305 = 46305, n = 1, r = 0.05.

Hence the time is obtained as follows:

[tex]40000(1.05)^t = 46305[/tex]

[tex](1.05)^t = \frac{46305}{40000}[/tex]

[tex]t = \log{\left(\frac{46305}{40000}\right)}{\log{1.05}}[/tex]

t = 3 years.

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When we use Gauss-Jordon elimination method, the final product is as follows

[tex]\left[\begin{array}{cccc}1&0&0&-1\\0&1&0&-3\\0&0&1&-2\end{array}\right][/tex]

The augmented matrix that represents a linear equation is as follow;

-5  7   7   -30

1    -4   4    3

2   -4   6    -2

Using Gauss-Jordon elimination method, we reduce every number to zero's and 1 except the values at the edge;

Divide row 1 by -5           Subtract row 1 from row 2

1   -7/5   -7/5   6                 1   -7/5    -7/5   6

1    -4      4      3                 0  -13/5   27/5  -3

2   -4      6     -2                 2    -4       6      -2

 R₃ - 2R₁                                    R₂ = -5R₂ ÷ 13

1    -7/5     -7/5     6                   1    -7/5     -7/5     6

0  -13/5    -27/5   -3                   0     1      -27/13   15/13

0   -6/5    44/5   -14                  0   -6/5    44/5   -14

R₁ + 7R₂ ÷ 5                                   R₃ + 6R₂÷5

1     0     -56/13  99/13                1     0     -56/13  99/13  

0     1    -27/13   15/13                 0     1    -27/13   15/13

0   -6/5    44/5   -14                   0     0    82/13    -164/13

   13R₃/ 82                                     R₁ + 56R₃/13

1     0    -56/13  99/13                  1     0     0        -1

0     1    -27/13   15/13                  0    1   -27/13  15/13  

0     0      1          -2                      0     0      1      -2

R₂ + 27R₃/13

1     0     0        -1

0     1     0        -3

0     0      1      -2

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Answer:

Step-by-step explanation:

132 + 106 + 81 + x = 360

319 + x = 360

x = 41

Answer:

y = 5x + 17

Step-by-step explanation:

gradient of perpendicular = - 1/ (-1/5) = - (-5) = 5.

equation of line, gradient 5, through (-2, 7) is

y - 7 = 5 (x - -2) = 5 (x + 2) = 5x + 10

y = 5x + 10 + 7

y = 5x + 17

Answer:

Part A

The amplitude of a cosine function is the distance between its highest and lowest points. In this case, the highest point is 8 feet and the lowest point is 8 - 12 = -4 feet, so the amplitude is 8 - (-4) = 12 feet.

The period of a cosine function is the time it takes for the function to complete one cycle of its movement. In this case, the bird completes one cycle of its movement every 16 seconds, so the period is 16 seconds.

Part B

The cosine function that could represent the situation is:

y = 6 cos(2πt / 16) + 8

This function has an amplitude of 6 feet and a period of 16 seconds. It also passes through the point (0, 8) when t = 0, which is the condition given in the problem.

Part C

The bird will reach its lowest height when the cosine function is equal to -6. This happens when 2πt / 16 = π / 2, or t = 4 seconds.

Here is a graph of the function:

[Image of a cosine function with an amplitude of 6 feet and a period of 16 seconds. The graph reaches its lowest point at t = 4 seconds.]

I hope this helps! Let me know if you have any other questions.

Step-by-step explanation:

Answer: 35, 48, 63

Step-by-step explanation:

This sequence is known as an arithmetic progression sequence.

      3 - 0 = 3

      8 - 3 = 5

      15 - 8 = 7

      24 - 15 = 9

Using this pattern, we will add 11 to 24 for the next term in the sequence. Then, we will continue this pattern for the next two terms.

➜ The pattern is resulting odd numbers from subtracting two terms in a row; 3, 5, 7, 9, [11], [13], [15].

      24 + 11 = 35

      35 + 13 = 48

      48 + 15 = 63

The correct statement is D. The inequality that represents the price, p, of each of the discounted tickets is p > 9.50.

Georgia wants to go to a concert.

The concert offers discounted tickets to groups that buy more than 12 tickets.

The price of each discounted ticket is less than the regular price of $9.50.

Now, let's break down each statement:

A. The inequality that represents the number of tickets, t, Georgia can buy at the discounted price is t > 12.

This statement is not necessarily true based on the information given. We know that the concert offers discounted tickets to groups that buy more than 12 tickets, but it doesn't provide any information about the maximum number of tickets Georgia can buy at the discounted price. The number of tickets Georgia can purchase could be more or less than 12.

B. The inequality that represents the number of tickets, t, Georgia can buy at the discounted price is t < 9.50.

This statement is incorrect. The inequality t < 9.50 doesn't make sense in this context because it implies that the number of tickets Georgia can buy is less than the price of a ticket, which is not relevant.

C. The inequality that represents the price, p, of each of the discounted tickets is p < 12.

This statement is incorrect as well. The inequality p < 12 suggests that the price of each discounted ticket is less than 12, but we don't have any information about the exact discount amount or the range of prices for the discounted tickets.

D. The inequality that represents the price, p, of each of the discounted tickets is p > 9.50.

This is the correct statement. We know that each discounted ticket costs less than the regular price of $9.50. Therefore, the price of each discounted ticket, represented by p, must be less than $9.50. Hence, the correct inequality is p < 9.50, or equivalently, p > 9.50.

In summary, the correct statement is D. The inequality that represents the price, p, of each of the discounted tickets is p > 9.50.

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If the spread of data set X is greater than the spread of data set Y, and the data sets are normally distributed. The statement that  is true is: C. The standard deviation of data set X is greater than the standard deviation of data set Y.

A data set standard deviation is often used to calculate its spread. As a result the standard deviation of data set X will be higher than the standard deviation of data set Y if the spread of data set X is bigger than the spread of data set Y.

The standard deviation may or may not have the same distribution as the the following :

Therefore the correct option is C.

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The integral of the given function is determined as (4²ˣ) / ln4    +  C.

option B.

The integral of the function is calculated by applying exponent rule of integration as follows;

The given integration ∫ 4²ˣ4⁻⁴

The given integral function is simplified as follows;

4²ˣ4⁻⁴ = ( 4² )ˣ 4⁻⁴ = 16ˣ4⁻⁴

The simplified expression is integrated as follows;

∫16ˣ4⁻⁴ = [ 16ˣ / ln 4 ] + C

where;

The integral of the given function is determined as;

∫ 4²ˣ4⁻⁴ = [ 16ˣ / ln 4 ] + C = (4²ˣ) / ln4    +  C

so option B is the correct answer.

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The value of x, y and z in the system equation is (1, 2, 3).

The solution of the equation can be determined by using Cramer's rule as follows;

[-5    -6      0] = [ -17]

[-3     -5     5]    [2   ]

[-6    -5      1]     [-13 ]

The determinant of the matrix is calculate as;

Δ = -5 (-5 + 25) + 6(-3 + 30) + 0(15 + 30)

Δ = 62

The x-determinant of the matrix is calculated as follows;

Δx = -17(-5 + 25) + 6(2 + 65) + 0

Δx = 62

The y-determinant of the matrix is calculated as follows;

Δy = -5(2 + 65) + 17(-3 + 30) + 0

Δy = 124

The z-determinant of the matrix is calculated as follows;

Δz = -5(65 + 10) + 6 (39 + 12) - 17(15 - 30)

Δz =  186

The value of x, y and z is calculated as follows;

x = Δx/Δ = 62/62 = 1

y = Δy/Δ = 124/62 = 2

z = Δz/Δ = 186/62 = 3

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The fully simplified product of (b minus 2 c)(negative 3 b + c) can be found by using the distributive property and combining like terms. The product is -3b^2 + 7bc - 2c^2, which has 3 terms.

Therefore, the statement "The simplified product has 2 terms" is false and the statement "The simplified product has 4 terms" is also false. The degree of each term is the sum of the exponents of the variables,

so the degree of the product is the highest degree among the terms. In this case, the highest degree is 2, which means that the statement "The simplified product has a degree of 2" is true and the statement "The simplified product has a degree of 4" is false.

Lastly, we can count the number of negative terms in the product to determine if the statement "The simplified product, in standard form, has exactly 2 negative terms" is true. We see that there are two negative terms, -3b^2 and -2c^2, so this statement is true.

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The result of the matrix 4|D| -3|B|

[tex]\left[\begin{array}{ccccc}31&6&36&-14&-57\\-19&-27&-18&18&27\\47&14&-1&-40&9\\10&12&-19&-15&-58\\-14&-23&10&-8&19\end{array}\right][/tex]

To solve the 5 x 5 matrix, we have to multiply and subtract the matrix this way;

4 x 7 - 3 x -1 = 31

4 x -6 - 3 x-10 = 6

4 x 3 - 3 x-8 = 36

4 x -8 - 3 x-6 = -14

4 x -9 - 3 x 7 = -57

Therefore 31, 6, 36, -14, and -57 makes the first role of the matrix 4|D| - 3|B|

4 x 2 - 3 x 9 = -19

4 x -9 - 3 x -3 = -27

4 x -6 - 3 x -2 = -18

4 x 6 - 3 x 2 = 18

4 x 9 - 3 x 3 = 27

Therefore -19, -27, -18, 18, and 27 make the second role of the matrix 4|D| - 3|B|

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Upon performing polynomial long division, we find that the width is (x - 7). So, the dimensions of the rectangle are length (x - 5) and width (x - 7).

Given that the area of a rectangle is equal to its length multiplied by its width, we can set up the equation as follows:

Area = Length × Width

In this case, the area is given as (x^2 - 12x + 35), and the length is (x - 5). We are asked to find the width.

So, we can rewrite the equation as:

(x^2 - 12x + 35) = (x - 5) × Width

To find the width, we need to divide the area by the length:

Width = (x^2 - 12x + 35) ÷ (x - 5)

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Answer:

47
Step-by-step explanation:

The length of side p is approximately 21.67

In triangle PQR, we are given the values of sin P, sin R, and the length of side r. We need to find the length of side p.

We can use the law of sines to solve this problem. The law of sines states that the ratio of the length of a side of a triangle to the sine of the opposite angle is constant. Mathematically, it can be represented as:

p / sin P = r / sin R

Plugging in the given values, we have:

p / 0.5 = 13 / 0.3

Cross-multiplying, we get:

0.3p = 13 * 0.5

0.3p = 6.5

Dividing both sides by 0.3, we find:

p ≈ 21.67

Therefore, the length of side p is approximately 21.67.

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