Suppose that the function g is defined, for all real numbers, as follows. Find g(-5) ,g (-2), and g(-1)

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 ratio of true positives to false positives, given the accuracy of the test would be 239 : 711 .

Here. we have,

First, find the number of people who have the flu:

= 5. 2 % x 10, 000

= 520 people

Those who don't have it:

= 10, 000 - 520

= 9, 480 people

True positive :

= 92 % x 520

= 478 people

False negatives:

= 8 % x 520

= 42 people

False positives:

= 15 % x 9, 480

= 1, 422 people

The ratio of true positives to false positives is therefore:

478 : 1, 422

239 : 711

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Answer:

The heights and shadows of the objects are legs of similar right triangles. We can use these similar triangles to set up a proportion to solve for the height of the building.

2.9 / 1.26 = x / 47.25

x = 108.75 m

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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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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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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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 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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1. The geometric sequence is B. -1, 2, -4, 8.

2. The sixth term is 8192.

3. The next three terms are 1, 1/2 and 1/4.

4. The 10th term is 4374.

5. The eight term is 1/16.

6. The next three terms in the geometric sequence are -1/36 1/216, and - 1/1296

7. The 8th term or the sequence will be 4.

8. The 6th term of the geometric sequence is 2048.

The 6th term of the sequence is:

a₆  = -2 * 4⁵ = -2 * 1024 = -2048

The common ratio of the sequence is:

r = 8/16 = 4/8 = 2/4 = 1/2

16 * (1/2) = 8

8 * (1/2) = 4

4 * (1/2) = 2

Therefore, the next three terms in the geometric sequence are 8, 4, and 2.

The common ratio of the sequence is:

r = 6/(-36) = -1/6

-1/6 * (1/6) = 1/36

1/36 * (1/6) = 1/216

-1/216 × 1/6 = -1 /1296

Therefore, the next three terms in the geometric sequence are -1/36, 1/216 and 1/1296.

The 8th term or the sequence will be:

= 512 × 0.5 × 0.5 × 0.5 × 0.5⁴

= 4

The 6th term of the geometric sequence is:

= 2 × 4 × 4 × 4 × 4 × 4

= 2048

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Answer: 18.75

Step-by-step explanation:

small greeting cards = 1.80

large greeting cards = 3.85

1.80 x 4 = 7.20

3.85 x 3 = 11.55

7.20 + 11.55 = 18.75

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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Answer:



Step-by-step explanation:

Answer:

24.6%

Step-by-step explanation:

denote B: bad mood; H: healthy; D: periodontal disease

P(B|D)

=P(B,D) / P(D)

=P(B,D) / [P(D,B)+P(D,H)]          ### law of total probability ###

=P(D|B)P(B) / [P(D|B)P(B) + P(D|H)P(H)]

=(0.85)*(0.1)/[(0.85)*(0.1) + (0.29)*(0.9)]

=0.245664739

Answer:

5.6 mi

Step-by-step explanation:

1 km is around 0.6 (estimate) miles so just multiply both sides by 9 and 9km = around 5.4 miles and 5.6 is the closest so it is most likely that

The square cake can be distributed in such a way that each person out of 15 guest may get 8  , with the least amount left over .

The Correct option is (1).

What is Volume of any shape?

The volume of an object is the amount of space occupied by the object or shape, which is in three-dimensional space. It is usually measured in terms ofof cubic units.

First find the volume of square cake,

Volume = l x b x h

= 8 x 8 x 2

=128

Now there are total 15 guests, Each of get

=

=  8.533

2.Volume of round cake (d= 6 inch, r=3 inch, h= 3 inch)

Volume=

=

=

=28.26  

Now there are total 15 guests, Each of get

=

= 1.884

3.Volume of round cake (d= 8 inch, r=4inch, h= 3 inch)

Volume=

=

=

=50.24  

Now there are total 15 guests, Each of get

=

=  3.349

4. Volume = l x b x h

= 6 x 9 x 2

=108

Now there are total 15 guests, Each of get

=

=  7.2

According to question each person may get 8 .

Only the square cake can be distributed in such a way that each person may get 8 , with the least amount left over .

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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Answer:

Step-by-step explanation:

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.

Answer:

Step-by-step explanation:

The margin of error represents the range of values within which the true population mean is likely to fall with a certain level of confidence. To calculate the range of values for the actual mean number of hours spent in meetings in a week, we can use the formula:

(actual mean) = (sample mean) ± (margin of error)

Substituting the given values, we get:

(actual mean) = 4.3 ± 1.1

To find the range of values, we need to compute the upper and lower bounds of this interval:

Lower bound = 4.3 - 1.1 = 3.2 hours

Upper bound = 4.3 + 1.1 = 5.4 hours

Therefore, the actual mean number of hours spent in meetings in a week is between 3.2 and 5.4 hours.

The model rocket will have a height of 30 meters at a time of 0.523 seconds and 11.477 seconds, respectively.

In this problem we have the case of a model rocket being launched, the height of the rocket in time is described by the following quadratic equation:

h(t) = 60 · t - 5 · t²

Where:

We need to determine the time associated to a given height of 30 meters above the ground. If we know that h(t) = 30, then the times are, respectively:

30 = 60 · t - 5 · t²

6 = 12 · t - t²

t² - 12 · t + 6 = 0

(t - 11.477) · (t - 0.523) = 0

t = 11.477 s. or t = 0.523 s.

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Answer: The odds against selecting a diamond are 3:1 (or 3/4)

Step-by-step explanation: Which means that there are 3 ways to not select a diamond and only 1 way to select a diamond. The odds in favor of selecting a diamond are 1:3 (or 1/4), which means that there is only 1 way to select a diamond and 3 ways to not select a diamond.

Therefore, 70% of state capitals do not have interstate highways serving them.

Percent is a way of expressing a number as a fraction of 100. The term "percent" means "per hundred". Percentages are usually denoted by the symbol %, which is placed after the numerical value. Percentages are used in many fields, including finance, science, and everyday life, to represent proportions, rates, and changes in quantities.

Here,

If all but 36 of the 120 state capitals have an interstate highway serving them, then the number of capitals that do not have interstates is 120 - 36 = 84.

To find the percentage of capitals that do not have interstates, we need to divide the number of capitals without interstates by the total number of capitals and then multiply by 100:

84 / 120 x 100% = 70%

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Answer:

240 8018

Step-by-step explanation:

two hundred forty eight thousand and eighteen

The likelihood that he or she has a pet given that a student does not have a sibling is 60%. Therefore, the correct option is option D.

Let A denote the event that a student does not have a sibling.

Let B denote the event that a student has a pet.

Then A∩B will denote the event that the student does not have a sibling but he has a pet.

Let P denote the probability of an event in this question.

Then, we have to find the conditional probability of event B which means the likelihood of event B occurring based on the occurrence of event A.

P(B|A)

We know that:

[tex]P(B|A) = \frac{P(A\cap B)}{P(A)}[/tex]

Also from the table, we have:

P(A)=0.25

and P(A∩B)=0.15

Hence,

[tex]P(B|A) = \frac{0.15}{0.25}[/tex]

[tex]P(B|A) = \frac{3}{5}[/tex]

[tex]P(B|A) = 0.6[/tex]

which in percentage is:

0.6 * 100 = 60 %

Therefore, the likelihood that he or she has a pet too is 60% and the correct option is option D.

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Note that the five number summary and range for the data are given as follows:

The five-number summary and range were calculated using the given stem and leaf plot. The minimum value is 102, the first quartile (Q1) is 226, the median (Q2) is 340, the third quartile (Q3) is 492, and the maximum value is 670. The range is the difference between the maximum and minimum values, which is 568.

The range is calculated by subtracting the minimum value from the maximum value:

Range: 607 - 102 = 505

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The total area of the figure with points A(-5,4), B(4,4), C(4,1),  D(1,1), E(4,-3) F(4,-6), G(-5, -6) has been found as  51 square units.

A graph is described as a mathematical representation of a network and it describes the relationship between lines and points.

The first step is to divide the figure is into two rectangles and a triangle.

we notice that the points B, C, E, and F will form a rectangle with base BC and height of 7 units.

We also notice that the  points A, D, and G form a rectangle with base AG and height 5 units.

we notice that  points C, D, and E form a triangle with base CE and height 4 units.

we notice that the  length of BC is 4 - 4 = 0, therefore  the area of the rectangle with base BC is 0.

The area of the rectangle with base AG is (4 - (-5)) x 5 = 45 square units. The area of the triangle with base CE is (4 - 1) x 4 / 2 = 6 square units.

In conclusion, the total area of the figure with points A(-5,4), B(4,4), C(4,1),  D(1,1), E(4,-3) F(4,-6), G(-5, -6)  is 45 + 6 = 51 square units.

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