how do i use change of base formula with logorothims

Answer:

A

Step-by-step explanation:

ita a bc i know its I did this before

The solution to the system of equations y=-3x+5 and 2x+y=6 is (x,y) = (-1,8).

To solve the system of equations using substitution, we can substitute the expression for y in the second equation with the expression for y in the first equation:

2x + y = 6

2x + (-3x + 5) = 6 (substitute y = -3x + 5)

2x - 3x + 5 = 6

-x + 5 = 6

-x = 1

x = -1

Now that we have found the value of x, we can substitute it back into either of the original equations to find the value of y:

y = -3x + 5

y = -3(-1) + 5

y = 8

We can check our solution by verifying that it satisfies both equations:

y = -3x + 5 => 8 = -3(-1) + 5 => 8 = 8 (true)

2x + y = 6 => 2(-1) + 8 = 6 => 6 = 6 (true)

Thus, our solution is correct.

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The value of variable y from the system of vertical angles is equal to 125.

In this problem we need to determine by algebra properties the value of a variable y from a system of two pairs of vertical angles. The system is represented by the following expression:

x + 20 = 3 · x - 50

2 · x = 70

x = 35

Then, by definition of supplementary angles:

(x + 20) + y = 180

55 + y = 180

y = 125

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The measure of angle a is solved to be 22 degrees

When two secant lines intersect outside a circle the formula for the vertex angle at a is

To find the vertex angle at point A where two secant lines intersect outside a circle, you can use the following formula:

angle A = 1/2 * (measure of arc 1 - measure of arc 2)

let arc 1 = 140 degrees

arc 2  = 96 degrees

plugging in the values

angle A = 1/2 * (140 - 96)

angle A = 1/2 * 44

angle A = 22 degrees

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The Volume of the resulting rectangular box is calculated as: 72 cubic units.

A rectangular box's volume can be calculated using the formula expressed below:

Volume = length * width * height.

Using the image of the net of the rectangular box, when folded, the dimensions of the box would be:

Length = 6 units

Width = 10 - 6 = 4 units

Height = 11 - 4 - 4 = 3 units

Plug in the values:

Volume of rectangular box = 6 * 4 * 3 = 72 cubic units.

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Answer: 55

Step-by-step explanation:

The scale of Sarah's zoomed-in map is 1cm = 1000 cm or 1cm = 10 m. So the answer is (B) 1cm = 10 m.

To find the scale of Sarah's zoomed-in map, we can use the ratio of the width of the park on the map to its actual width.

Let's first convert the width of the park from meters to centimeters, since the width of the park on the map is given in centimeters.

15 m = 1500 cm

Next, we can set up a proportion:

0.5 cm / x = 1500 cm / (3x)

where x is the scale of the zoomed-in map.

Simplifying the proportion, we get:

0.5(3x) = 1500

1.5x = 1500

x = 1000

Therefore, the answer is (B) 1cm = 10 m.

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The statement that gives the best approximation of the solutions of the equation f(x) = g(x) is:

option A. The solutions are where the graphs of the functions intersect at ≈ 1 and ≈ 3.642.

To find the solutions of the equation f(x) = g(x), set f(x) = g(x) and then solve for x. Using the given functions, we have:

log(x - 1) = (⅓x² - 4)

Exponentiating both sides:

x - 1 = e^(⅓x²-4)

Solve for x:

x = e^(⅓x²-4) + 1

Therefore, the solutions to the equation f(x) = g(x) are given by the values of x for which the graphs of the functions intersect. To find these points of intersection, plot the graphs of the functions and look for the points where they intersect. In other way round, solving for x numerically:

Based on the given functions, it is clear that g(x) is an upward-facing parabola with its vertex at (0, -4) and x-intercepts at (-2√3, 0) and (2√3, 0). The function f(x) is defined only for x > 1, and its graph is a monotonically increasing curve that approaches infinity as x approaches 1 from the right.

Therefore, there is only one point of intersection between the graphs of f(x) and g(x), which occurs to the right of x = 1. This point of intersection can be found numerically using a calculator or computer software, and it is approximately x ≈ 3.642.

Thus, the statement that gives the best approximation of the solutions of the equation f(x) = g(x) is:

OA. The solutions are where the graphs of the functions intersect at ≈ 1 and ≈ 3.642.

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The area of the composite figure is equal to 75.5 square meters

The composite figure can be observed to be made up of a big rectangle, a smaller rectangle, a triangle and a smaller triangle. We calculate for the area of the four shapes and sum the results to get the total area of the composite figure as follows:

area of the big rectangle = 6 m × 4 m = 24 m²

area of the smaller rectangle = 7 m × 2 m = 14 m²

area of the big triangle = 1/2 × 10 m × 6 m = 30 m²

area of the smaller triangle = 1/2 × 5 m × 3 m = 7.5 m²

total area of the composite figure = 24 m² + 14 m² + 30 m² + 7.5 m²

total area of the composite figure = 75.5 m²

Therefore, the area of the composite figure is equal to 75.5 square meters

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Answer:

D.

x = 31; m∠ROS = 62°

Step-by-step explanation:

we know that a right angle is always 90 degrees, so we subtract 90° by ∠QOR:

[tex]90 - 28 = 62[/tex]

Then we insert the 62 with the (2x)

[tex]62 = 2x[/tex]

we divide both sides by 2 and we get:

[tex]31 = x[/tex]

so the answer is D

x is equal to 31 and we can multiply 31 by 2 and we can get 62°

The given Mayan number which is: :|| :|| in decimal (base 10) is 42.

The Mayan number system is a base-20 system that uses three symbols: a dot (.), a horizontal bar (|), and a shell-like symbol (:). The symbol for zero is a shell-like symbol (:).

In order to obtain the decimal equivalent (base 10) of a Mayan numeral, it is imperative to comprehend the positional significance of each constituent symbol.

The numerical system employed here assigns a weight to each digit that corresponds to a certain power of 20. Specifically, the weight associated with the least significant digit is 20 raised to the exponent of 0, while the weight of the digit to its immediate left is 20 raised to the exponent of 1, and so forth.

The Mayan number given, : :|| :||, can be broken down as follows:

The leftmost position has no symbol, which represents a value of 0.

The second position from the left has two shell-like symbols (:), which represent a value of 2x20¹ = 40.

The third position from the left has two horizontal bars (||), which represent a value of 2x20⁰ = 2.

The given Mayan number in decimal (base 10) is 40 + 2 = 42.

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The equation of the tangent line to the curve [tex]x^\((2/3) + y^\((2/3)[/tex]= 10 at the point (-27, 1) is  -(1/3)(x + 27).

First step is to take the  derivative of both sides of the equation

[tex](2/3 )x ^\((-1/3) + (2/3)y ^\((-1/3)*dy/dx = 0[/tex]

Solve for dy/dx

[tex]dy / dx = -(y ^\(1/3)/x ^(1/3))[/tex]

Substitution

[tex]dy /dx = -(1 ^\((1/3)/(-27) ^ \((1/3)) = -(1/3)[/tex]

Formulate a linear equation

y - y1 = m(x - x1)

Where:

m= slope

(x1, y1)= point (-27, 1):

So,

y - 1 = -(1/3)(x + 27)

Therefore the equation is y - 1 = -(1/3)(x + 27).

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The perimeter of the given triangle is 12.4x-8.0y+5.8z. Therefore, option C is the correct answer.

Given that, triangle has sides of lengths of (6.2x-9.4y) centimeters, (6.2x-2.5z) centimeters and (8.3z+1.4y) centimeters.

We know that, perimeter of triangle is sum of all the sides.

Now, perimeter = 6.2x-9.4y+6.2x-2.5z+8.3z+1.4y

= (6.2x+6.2x)+(-9.4y+1.4y)+(-2.5z+8.3z)

= 12.4x-8.0y+5.8z

Therefore, option C is the correct answer.

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The force parallel to the incline that would be required to hold the monolith on this causeway is 32, 120.52 Newtons.

The formula for finding the force parallel to the incline:

= mass of the monolith x acceleration due to gravity x sin( slope angle )

Mass in kg :

=  57 tons x 1000

= 57, 000 kg

Force parallel to the incline:

= 57, 000 kg x 9. 81 m/s² * sin ( 1.4 degrees )

= 57, 000 kg x 9.81 m/s² x sin ( 0.0244 radians )

= 32, 120.52 Newtons

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The price of the bed after mark up is $475.00

Discount results in the reduction of the selling price of the product, which makes it more attractive for the customer.

The first discount given is 25% , therefore the price of the bed before mark up is

25/100 × 800

= 25×8

= 200

the price before mark up = 800-200 = $600

Another 25% is given after the first sale, there the price of the bed after mark up is

25/100 × 600

=$ 125

price after markup = 600-125

= $475.00

therefore the price of the bed after markup is $475.00

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Answer:

Step-by-step explanation:

m = n + 1 + x/p

First, we can start by subtracting (n+1) from both sides:

m - (n+1) = x/p

Then, we can multiply both sides by p to isolate x:

p(m - (n+1)) = x

So the final answer is:

x = p(m - (n+1))

The domain of the 1st piece is -10 < x < 0, for 2nd piece is 0 < x < 4, and for 3rd piece is 4 < x < 8.

A function is a particular kind of relationship where there is a set domain and range, and every input value in the domain is associated with a single output value in the range.

We have a piecewise function shown in the picture:

From the graph:

f(x) = -2, Domain: -10 < x < 0

f(x) = 2x + 1, Domain: 0 < x < 4

f(x) = (-1/2)x, Domaim: 4 < x < 8

Thus, the domain of the 1st piece is -10 < x < 0, for 2nd piece is 0 < x < 4, and for 3rd piece is 4 < x < 8.

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Answer:

24

Step-by-step explanation:

10^2+x^2=26^2

x^2=26^2-10^2

x^2=576

x=[tex]\sqrt{576}[/tex]=24

The image coordinates of NMO after the translation is option d: N′(−5, −6), M′(−2, −5), and O′(−3, −7).

To get the image coordinates of the preimage translated -2 units to the left, we simply subtract -2  from the y-coordinates of each vertex:

N' = (Nx - (-2), Ny) = (−5 , 2- (-2)) = (−5, 4)

M' = (Mx - (-2), My) = (−2 , 1 - (-2)) = (−2, 3)

O' = (Ox - (-2), Oy) = (−3 , 3- (-2)) = (−3, 5)

Therefore, based on the above, the image coordinates of NMO after the translation are N′(−5, 4), M′(-2,3 ), O′(−3, 5)

So the reflected vertices are:

The distance from N to the line y = -2 is 4, and -6 - (-2) = -4, so one need to move down 4 units to have  a y-coordinate of -6.

The distance from M to the line y = -2 is 3, and -5 - (-2) = -3, so  one need to move down 3 units to have a y-coordinate of -5.

The distance from O to the line y = -2 is 5, and -7 - (-2) = -5, so  one need to move down 5 units to have  a y-coordinate of -7.

So it will be: N′ (−5, −6), M′(−2, −5), O′(−3, −7)

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Answer:

---------------------

Given triangle MNO with vertices:

It is reflected in line y = - 2.

This reflection doesn't affect the x-coordinates of the vertices and the y-coordinates change.

Since y = - 2 is the line of symmetry,  it also represents midpoints of the segments formed by corresponding endpoints of image and preimage.

Using midpoint equation, find the y-coordinates.

So the coordinates of the image are:

This is matching the option D.

See attached for visual representation of the problem.

The matrix formed by performing R2 -> 4R1 + R2 on M is given as follows:

[tex]\left[\begin{array}{ccc}-4&3&1\\-18&9&8\end{array}\right][/tex]

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

[tex]\left[\begin{array}{ccc}-4&3&1\\-2&-3&4\end{array}\right][/tex]

The rows are given as follows:

Hence the row 2 of the resulting matrix has the elements given as follows:

Thus the resulting matrix is:

[tex]\left[\begin{array}{ccc}-4&3&1\\-18&9&8\end{array}\right][/tex]

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Answer:

x = 16.2

Step-by-step explanation:

Using trig functions:

cos theta = adjacent side/ hypotenuse

cos 72 = 5/x

Solving for x

x = 5 /cos 72

x=16.18033

Rounding to the nearest tenth

x = 16.2

Answer: x = 16.2 units

Step-by-step explanation:

     We are given an angle (72°), the adjacent side (5), and the hypotenuse (x). We will use the cosine function to solve.

cos(θ) = [tex]\frac{adjacent}{hypotenuse }[/tex]

cos(72°) = [tex]\frac{5}{x }[/tex]

xcos(72°) = 5

x = [tex]\frac{5}{cos(72\°)}[/tex]

x = 16.1803398 ≈ 16.2

Answer:

A and D

Step-by-step explanation:

A- when multiplying -5 and -3 it should be positive 15.

D- when dividing 3 they just outta nowhere forgot about the negative, it should've been -5

The result of adding -4 (row 1) to row 3 is determined as (0  - 9   2)|12.

The resultant of the row multiplication in the Matrice is calculated by applying the following method;

row 1 in the given matrices = [1  2  1] | -5

To multiply row by -4, we will multiply each entity by 4 as shown below;

= -4(1 2  1) | -5

= (-4  -8  - 4) |20

To add the result to 3;

(-4  -8  - 4)|20 + (4  - 1   6) | -8

= (0  - 9   2)|12

Thus, the result of the row multiplication is determined by multiplying each entry in row 1, by - 4.

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The statement which correct describe "confidence-interval" is (d) confidence level is considered as probability that something will fall within "confidence-interval".

A "Confidence-Interval" is a statistical term used to describe the amount of uncertainty associated with a sample estimate. It refers to the percentage of all possible samples that can be expected to include the true population parameter being estimated.

For example, if a study reports a 95% confidence level, it means that 95% of all possible samples would contain the true population parameter. This can be considered as the probability that the estimate falls within a certain range or interval, called the confidence interval.

Therefore, option (d) is the correct statement.

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The given question is incomplete, the complete question is

Which statement correctly describes confidence level?

(a) A confidence level is the probability that a simple event will occur.

(b) A confidence level is a value that is added and subtracted from a given value to give an interval where something will likely happen.

(c) A confidence level is the probability that something will fall outside of a confidence interval.

(d) A confidence level is the probability that something will fall within a confidence interval.

Answer:

A confidence level is the probability that something will fall within a confidence interval.

Step-by-step explanation:

Plato/Edmentum

The calculated area of the triangle is 7.5 square inches

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

The triangle with the following dimensions

base = 6 inches

height = 2.5 inches

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

Area = 1/2 * base * height

Substitute the known values in the above equation, so, we have the following representation

Area = 1/2 * 6 * 2.5

Evaluate

Area = 7.5

Hence, the area of the triangle is 7.5 square inches

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The inverse of matrix C is C⁻¹ = [1/9 4/27] [1/27 -5/27].

To find the inverse of matrix C, use the following formula:

C⁻¹ = 1/det(C) × adj(C)

where det(C) = determinant of matrix C and adj(C) = adjugate of matrix C.

To find the determinant of C, use the following formula for a 2x2 matrix:

|a b|

|c d| = ad - bc

So:

det(C) = |5 1|

|12 -3| = (5)(-3) - (1)(12) = -27

Find the adjugate of C, which is the transpose of the matrix of cofactors of C, which is:

|-3|

| 1| = -(-3) = 3

So the upper left cofactor is 3, which is:

|12|

|-3| = -(12) = -12

So the upper right cofactor is -12, which is:

| 1|

|-3| = -(1) = -1

So the lower left cofactor is -1, which is:

|5|

|1| = 5

So the lower right cofactor is 5. Putting the cofactors into a matrix and transposing it gives the adjugate of C:

adj(C) = [ 3 -12]

[-1 5]

Now find the inverse of C:

C⁻¹ = 1/det(C) × adj(C)

= -1/27 × [ 3 -12]

[-1 5]

= [1/9 4/27]

[1/27 -5/27]

So the inverse of matrix C is:

C⁻¹ = [1/9 4/27]

[1/27 -5/27]

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The equation of the line AB,with slope of -1/2 is  y = (-1/2)x + 7/2. We then use the dot product to show that the direction vectors of the lines AB and AC are orthogonal, and the coordinates of point C is (6,7).

In an orthonormal frame (o; i; j), the points A and B are given as A(3;1) and B(1;2), and the line (AC) is given by the equation 2x-y-5=0.

To determine the equation of line (AB), we need to find the slope (or gradient) of the line passing through A and B. We can find this by using the formula

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

where (x1, y1) and (x2, y2) are the coordinates of two points on the line. Substituting the values of A and B, we get

slope = (2 - 1) / (1 - 3) = -1/2

Now that we have the slope, we can use the point-slope form of a line to find the equation of line (AB). Let (x,y) be any point on the line, then we have

y - 1 = (-1/2)(x - 3)

Simplifying, we get

y = (-1/2)x + 7/2

So the equation of line (AB) is y = (-1/2)x + 7/2.

To show that the direction vectors AB and AC of the lines (AB) and (AC) are orthogonal, we need to find the direction vectors of these lines and show that their dot product is zero. The direction vector of a line is just the vector that "points" in the direction of the line. To find it, we can take any two points on the line and subtract their coordinates to get a vector.

For line (AC), we can take A and C as two points on the line. Since we don't know the coordinates of C yet, we can solve the equation of line (AC) for x to get

x = (y + 5) / 2

Now substituting x by the above expression in the coordinates of point A we get

C((y + 5)/2, y)

Thus, the direction vector of line (AC) is

AC = C - A = ((y + 5)/2 - 3, y - 1) = (y/2 - 7/2, y - 1)

Similarly, we can take A and B as two points on the line (AB) to get the direction vector of line (AB):

AB = B - A = (1-3, 2-1) = (-2, 1)

Now, the dot product of AB and AC is

AB · AC = (-2)(y/2 - 7/2) + (1)(y - 1) = -y + 7

We can see that the dot product is zero when y=7. Hence, direction vectors AB and AC are orthogonal at point C(3,7).

To determine the coordinates of C, we substitute y=7 in the equation of line (AC) we obtained earlier

2x - y - 5 = 0

2x - 7 - 5 = 0

2x = 12

x = 6

Therefore, C has coordinates (6,7).

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The given scenario is Biased, since those who chose to respond may be systematically different than those who chose not to respond. So, the correct answer is C).

The sampling method is biased, since those who chose to respond may be systematically different than those who chose not to respond. This is known as non-response bias, where the opinions and characteristics of those who choose not to respond to the survey may differ from those who choose to respond.

Additionally, only one higher education institution per state was selected, which may not be representative of all institutions within the state. Therefore, the sample may not accurately reflect the population of students in the United States who pay for their higher education. So, the correct option is C).

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--The given question is incomplete, the complete question is given

" To answer the question, "How do students in the United States pay for their higher education?", a questionnaire was sent out to collect 100 responses from one higher education institution per state.

Is the sampling method biased or unbiased and why?

Unbiased, since a random sample was selected

Unbiased, since a large sample was selected

Biased, since those who chose to respond may be systematically different than those who chose not to respond

Biased, since all higher education institutions are represented

Unbiased, since a questionnaire was used

Biased, since every state will be represented in the sample"--