As as you approach zero from the left on a number line the integers ____ , but the absolute values of those integers ___?

The solutions for x between 0 and 2π are: x = π/2, (5π)/6, and (11π)/6.

To solve the equation 5 + sin(3x) = 4 for x on the unit circle, where x is between 0 and 2π, follow these steps:

1. Subtract 5 from both sides: sin(3x) = -1
2. Determine the angle for which sin is -1: sin(3x) = sin(3π/2)
3. Since the sine function has a period of 2π, the general solution is: 3x = 3π/2 + 2πk, where k is an integer.
4. Divide both sides by 3: x = π/2 + (2πk)/3

Now, find the values of x between 0 and 2π by trying different integer values of k:

- If k = 0, x = π/2
- If k = 1, x = π/2 + 2π/3 = (5π)/6
- If k = 2, x = π/2 + 4π/3 = (11π)/6

Thus, the solutions for x between 0 and 2π are: x = π/2, (5π)/6, and (11π)/6.

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The greatest of the three interior angles in the triangle is approximately 71.2 degrees.

To find out which of the three interior angles in the triangle is the greatest, we can use the fact that the largest angle is opposite the longest side. So, in this case, the longest side is 22, which means that the angle opposite it must be the greatest. We can use the Law of Cosines to find the measure of this angle:

c^2 = a^2 + b^2 - 2ab*cos(C)

where c is the length of the side opposite angle C, and a and b are the lengths of the other two sides. Plugging in the values we know, we get:

22^2 = 12^2 + 17^2 - 2*12*17*cos(C)

484 = 144 + 289 - 408*cos(C)

51 = 408*cos(C)

cos(C) = 51/408

C = cos^-1(51/408)

C ≈ 71.2 degrees

So the greatest of the three interior angles in the triangle is approximately 71.2 degrees.

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The lateral surface area of the rectangular prism is given as follows:

L = 60 cm².

The lateral surface area of a rectangular prism of length l, width w and height h is given by the equation presented as follows:

L = 2 ( l + w ) h

The dimensions for this problem are given as follows:

l = 3 cm, w = 2 cm and h = 6 cm.

Hence the lateral surface area of the rectangular prism is given as follows:

L = 2 x (2 + 3) x 6

L = 60 cm².

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Out of the 150 new couples, we can expect about:

45 * (150/240) = 28.125 couples where the husband prefers action but the wife does not.

30 * (150/240) = 18.75 couples where the wife prefers action but the husband does not.

What is probability?

Probability is a measure of the likelihood of an event occurring.

Based on the given data from the survey of 225 couples, we can construct a contingency table as follows:

                      Husband   Wife   Total

Action                45          30        75

Comedy            30          45        75

Drama               45          45        90

Total                 120        120       240

From the contingency table, we can see that:

Out of 240 respondents, 75 (45 from husbands and 30 from wives) preferred action movies.

Out of 240 respondents, 60 (30 from husbands and 30 from wives) preferred comedy movies.

Out of 240 respondents, 90 (45 from husbands and 45 from wives) preferred drama movies.

To answer the question of how many of the 150 couples will have exactly one spouse who prefers action movie, we can use the information that:

Out of 240 respondents, 45 husbands preferred action movies but their wives did not.

Out of 240 respondents, 30 wives preferred action movies but their husbands did not.

Therefore, out of the 150 new couples, we can expect about:

45 * (150/240) = 28.125 couples where the husband prefers action but the wife does not.

30 * (150/240) = 18.75 couples where the wife prefers action but the husband does not.

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The interquartile range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1) of the data. From the box-and-whisker plot given, the IQR is 12.

The box-and-whisker plot provides us with the following information:

Therefore, the range of the data is the difference between the maximum and minimum values:

Range = maximum value - minimum value = 13 - 1 = 12

So, the range of the data is 12.

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Full Question: The box-and-whisker plot shows the number of pigeons spotted by visitors at the park during the last weekend. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 What is the interquartile range of the data? Provide your answer below:

Image attached

Both situations can be represented by the number line as they both involve values either greater than or less than 75.

The number line the teacher drew can represent both Brooke's and Eileen's situations.

In Brooke's situation, the number line can represent the amount of money Christopher needs to earn to buy a new bicycle. If he needs to earn more than $75, any point on the number line greater than 75 would represent the amount of money he has earned that is sufficient for purchasing the bicycle.

In Eileen's situation, the number line can represent the time left in Paul's flight. If Paul has less than 75 minutes left, any point on the number line less than 75 would represent the time remaining in his flight.

Both situations can be represented by the number line as they both involve values either greater than or less than 75.

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(i) A Venn-diagram of this information is shown below.

(ii) The number of students who like picnic = 34

(iii) The number of students who like hiking only = 51

(iv) The percentage of students who like picnic only. = 45.33%

Let us assume that A represents the set of students who like picnic.

B represents the set of students who like the hiking.

The total number of students in a  class are: n(A U B) = 75

Out of 75 students, 10 like both the activities.

n(A ∩ B) = 10

The ratio of the number of students who like picnic to those who like hiking is 2 : 3

Let number of students like tea n(A) = 2x

and the number of students like coffee n(B) = 3x

n(A U B) = n(A) + n(B) - n(A ∩ B)

75 = 2x + 3x - 10

75 + 10 = 5x

85/5= x

x = 17

The number of students like picnic = 2x

                                                          = 2 × 17

                                                          = 34

The number of students like hiking = 3x

                                                           = 3 × 17

                                                           = 51

This informtaion in Venn diagram is shown below.

The percentage of students who like picnic only would be,

(34/75) × 100 = 45.33%

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The probability that the mean cost of the 9 weddings is more than the mean cost of the 6 showers is approximately 0.0207 or 2.07%.

The probability that the mean cost of the 9 weddings is more than the mean cost of the 6 showers can be found using the Z-score and the difference between the means of two normally distributed variables.

1: Calculate the difference in means and standard deviations.
Δμ = μ_weddings - μ_showers = $82.30 - $65 = $17.30
Δσ = sqrt((σ_weddings²/n_weddings) + (σ_showers²/n_showers)) = sqrt((18.20²/9) + (17.73²/6)) = $8.47

2: Calculate the Z-score.
Z = (Δμ - 0) / Δσ = (17.30 - 0) / 8.47 ≈ 2.04

3: Determine the probability using a Z-table.
P(Z > 2.04) ≈ 0.0207

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The differential equation that models the problem is: dN/dt = 0.2*N(t) - 3.04174E+07.

Let's denote the current number of fish in the pond by N(t), where t is time in years.

The rate of change of N(t) is given by the difference between the number of births and deaths, minus the number of fish harvested from the pond:

dN/dt = (0.6N(t)) - (0.4N(t)) - (3.04174E+07)

The first term represents the number of births, which is 60% of the current population N(t). The second term represents the number of deaths, which is 40% of the current population N(t). The third term represents the number of fish harvested from the pond each year.

Therefore, the differential equation that models the problem is:

dN/dt = 0.2*N(t) - 3.04174E+07

Note that we have simplified the expression (0.6-0.4)N(t) to 0.2*N(t) for simplicity.

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The combination of side lengths that would not form a triangle is C.XY = 11 mm, YZ = 14 mm, XZ = 21 mm.

We shall use the triangle inequality theorem to determine if a set of side lengths can form a triangle.

The triangle inequality theorem says that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

We shall calculate each of the options:

For option A:

XY + YZ = 7 mm + 14 mm = 21 mm which is < XZ = 25 mm.

Therefore, option A does form a triangle.

For option B:

XY + YZ = 11 mm + 18 mm = 29 mm, which is > XZ = 21 mm.

YZ + XZ = 18 mm + 21 mm = 39 mm, which is > XY = 11 mm.

XY + XZ = 11 mm + 21 mm = 32 mm, which is > YZ = 18 mm.

Therefore, option B does form a triangle.

For option C:

XY + YZ = 11 mm + 14 mm = 25 mm, and is > XZ = 21 mm.

Therefore, option C does not form a triangle.

For option D:

XY + YZ = 7 mm + 14 mm = 21 mm, which is > XZ = 17 mm.

YZ + XZ = 14 mm + 17 mm = 31 mm, which is > XY = 7 mm.

XY + XZ = 7 mm + 17 mm = 24 mm, which is > YZ = 14 mm.

Therefore, option D does form a triangle.

Therefore, the combination of side lengths that would not form a triangle is XY = 11 mm, YZ = 14 mm, XZ = 21 mm.

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Answer:

The approximated measure of this angle is 90°, so this may be a right angle.

Place a sheet of paper so that the corner corresponds to the angle. You will notice that the lines will closely align with the edges of the paper.

The toys approach each other at the distance of  630 cm.

To solve the problem, we can use the Pythagorean theorem.

Let the distance between the tugboat and the sail ship be d, and

let t be the time in seconds since they started moving.

Then we have:

Distance traveled by the tugboat (in cm) = 5t

Distance traveled by the sail ship (in cm) = 7t/sqrt(2)

Using the Pythagorean theorem, we have:

d² = (5t)² + (7t/(\sqrt(2)))²

d² = 25t² + 24.5t²

d² = 49.5t²

d = \sqrt(49.5)t

To find how closely the two toys approach each other, we need to find the minimum value of d.

This occurs when t is maximized, which happens when the toys are closest to each other.

The sail ship travels a distance of 8\sqrt(2) meters in the Northeast direction, which is equivalent to 800\sqrt(2) cm. Therefore, the time taken for the sail ship to travel this distance is:

t = (800\sqrt(2) cm) / (7 cm/(\sqrt(2))) = 200\sqrt(2) seconds

Substituting this value of t in the equation for d, we get:

d = \sqrt(49.5)(200\sqrt(2)) = 630 cm (corrected)

Therefore, the minimum distance between the two toys is 630 cm.

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The equation that could represent each of the graphed polynomial function include the following:

First graph: y = x(x + 2)(x - 3)

Second graph: y = x⁴ - 5x² + 4

In Mathematics and Geometry, a polynomial graph simply refers to a type of graph that touches the x-axis at zeros, roots, solutions, x-intercepts, and factors with even multiplicities.

Generally speaking, the zero of a polynomial function simply refers to a point where it crosses or cuts the x-axis of a graph.

By critically observing the graph of the polynomial function shown in the image attached above, we can logically deduce that the first graph has a zero of multiplicity 1 at x = 2 and zero of multiplicity 1 at x = -3.

Similarly, we can logically deduce that the second graph has a zero of multiplicity 2 at x = 2 and zero of multiplicity 2 at x = -2.

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Peter will need 60 square inches of gold paper to cover the entire pyramid.

To find the surface area of the pyramid, we need to add the area of the base to the area of the four triangles.

Area of the base = 20 square inches

Area of each triangle = 10 square inches

Total area of the four triangles = 4 x 10 = 40 square inches

Total surface area = Area of base + Total area of four triangles

                                = 20 + 40

                                = 60 square inches

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In this scenario, the dependent variable is "how many jumps I can do," while the independent variable is "how long I am jumping (2 minutes)."

The relationship between these variables can be described as follows: The number of jumps completed depends on the duration of time spent jumping, with a specific focus on a 2-minute interval.

When we say the dependent variable is "how many jumps I can do," it means that the number of jumps completed is determined by or depends on the independent variable, which is the duration of time spent jumping.

This suggests that as the duration of time increases or decreases, it will likely have an impact on the number of jumps performed.

In this particular case, you have specified a 2-minute interval as the focus. It suggests that you are examining the relationship between the number of jumps completed and the specific duration of 2 minutes.

This implies that you are interested in understanding how the number of jumps varies within this fixed time frame.

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A linear function would be the  best fit for the data.

A function that would be the best for this data is: D. y = -4/25(x) + 10

The amount of snow that would be on the ground when the temperature reaches 55° is 1.2 inches.

In this scenario, the temperature would be plotted on the x-axis (x-coordinate) of the scatter plot while the snow (inches) would be plotted on the y-axis (y-coordinate) of the scatter plot through the use of Microsoft Excel.

On the Microsoft Excel worksheet, you should right click on any data point on the scatter plot, select format trend line, and then tick the box to display an equation for the line of best fit (trend line) on the scatter plot.

From the scatter plot (see attachment) which models the relationship between the temperature and the snow (inches), an equation for the line of best fit is given by:

y = -0.16x + 10

y = -4/25(x) + 10

When x = 55, the amount of snow is given by;

y = -4/25(55) + 10

y = 1.2 inches.

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The loss price of the product is Rs 11. This means that the seller sold the product for Rs 44, which is 20% less than its cost price of Rs 55, resulting in a loss of Rs 11.

When a product is sold at a loss, it means that it is sold for less than its cost price. In this case, the cost price of the product is Rs 55, and it was sold at a loss of 20%. This means that the selling price of the product is 80% of its cost price. To find out the selling price, we can multiply the cost price by 80% or 0.8.

Selling price = Cost price x (100% - Loss%)

Selling price = Rs 55 x (100% - 20%)

Selling price = Rs 55 x 80%

Selling price = Rs 44

So, the selling price of the product is Rs 44. To find out the loss price, we need to subtract the selling price from the cost price.

Loss price = Cost price - Selling price

Loss price = Rs 55 - Rs 44

Loss price = Rs 11

Therefore, the loss price of the product is Rs 11.

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The measure of angle B is 60 degrees. The given triangle ABC is an isosceles triangle since two sides, AC and BC, are equal in length to 7 cm.

Therefore, the angle opposite the base (AB) will be equal in measure.

To find the measure of angle B, we need to use the cosine rule, which relates the length of sides of a triangle to the cosine of the angle opposite the side.

According to the cosine rule, cos(B) = ([tex]a^{2}[/tex] + [tex]c^{2}[/tex] - [tex]b^{2}[/tex]/(2ac). Substituting the values, we get cos(B) = ([tex]7^{2}[/tex] + [tex]7^{2}[/tex] - [tex]7^{2}[/tex])/(2x7x7), cos(B) = 1/2, B = [tex]cos^{-1}[/tex](1/2), B = 60°

Therefore, the measure of angle B is 60 degrees.

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Answer:

1) a = 14

2) -4 (x - 2)² - 5

Step-by-step explanation:

To obtain a vertex, you take h and k in a equation.

So a(x-h)²+k = a(x-2)² -5

For the point (1, - 9),

a[(1)-2]² - 5 = - 9

a(1) = -9+5

a = -4

so the final equation is

-4(x-2)² - 5

I'm not 100% sure about this but I tried. Let me know if it makes sense

[tex]~~~~~~~~~~~~\textit{distance between 2 points} \\\\ (\stackrel{x_1}{-3}~,~\stackrel{y_1}{-6})\qquad (\stackrel{x_2}{5}~,~\stackrel{y_2}{0})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ d=\sqrt{(~~5 - (-3)~~)^2 + (~~0 - (-6)~~)^2} \implies d=\sqrt{(5 +3)^2 + (0 +6)^2} \\\\\\ d=\sqrt{( 8 )^2 + ( 6 )^2} \implies d=\sqrt{ 64 + 36 } \implies d=\sqrt{ 100 }\implies d=10[/tex]

The value of s is, 27 yards

:: Volume of cube with side s, is equal to s³

So, as the given volume is 19,683 cubic yards.

Therefore, it can related as,

s³ = 19,683 (yards)³

So,

s = ∛(19,683) yards

s = 27 yards

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Thus, the coordinates of Quadrilateral A'B'C'D' after the dilation with the scale factor of 1/2 are -  A'(1.5, 2), B'(0.5, 5), C'(6, 7), D'(4.5, 1.4).

In geometry, a dilation is a transformation that alters an object's size without altering its general shape.

Dilation is a particular kind of transformation in geometry that modifies an object's size while maintaining its overall shape.

Given:

scale factor = 1/2

coordinates of Quadrilateral ABCD

A(3,4) , B(1,10) ,C(12,14), D(9,3)

Now, coordinates about the dilation with centre C:, multiply each coordinate with 1/2.

A'(3*1/2,4*1/2)   --> A'(1.5, 2)

B'(1*1/2,10*1/2)    ---> B'(0.5, 5)

C'(12*1/2,14*1/2), --> C'(6, 7)

D'(9*1/2,3*1/2) ---> D'(4.5, 1.4)

Thus, the coordinates of Quadrilateral A'B'C'D' after the dilation with the scale factor of 1/2 are -  A'(1.5, 2), B'(0.5, 5), C'(6, 7), D'(4.5, 1.4).

Graph is attached.

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The rate of appreciation of the classic car is 18% per year.

An exponent is a mathematical operation that indicates how many times a number or expression is multiplied by itself. It is represented by a superscript number that is written to the right and above the base number or expression. The exponent tells us how many times the base is multiplied by itself.

The value V of the classic car appreciates exponentially, and it is represented by the formula:

V = 32,000[tex]1.18^{2}[/tex]

The term [tex]1.18^{t}[/tex] represents the factor by which the value of the car increases each year. If we calculate this factor for one year (t=1), we get:

(1.18)¹= 1.18

This means that the value of the car increases by 18% in the first year. Similarly, if we calculate the factor for two years (t=2), we get:

(1.18)² = 1.39

This means that the value of the car increases by 39% in the first two years (18% in the first year and an additional 21% in the second year).

Therefore, the rate of appreciation of the classic car is 18% per year.

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To determine if two lines intersect, compare their slopes and y-intercepts, and solve their equations simultaneously.

To determine if two lines intersect, you need to compare their key features, such as their slope and y-intercept.

If the slopes of the two lines are different, then they will intersect at some point.

If the slopes are the same, then the lines may or may not intersect, depending on whether or not their y-intercepts are also the same.

To find the point of intersection, you can set the two linear functions equal to each other and solve for the variable. The resulting value will give you the x-coordinate of the intersection point, which can then be substituted back into either equation to find the corresponding y-coordinate.

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There are 3 ordered pairs (x, y) that satisfy the equation y=(x-5)^2-1.

To find the ordered pairs (x, y) for 3 < x < 7 that satisfy the equation y=(x-5)^2-1, follow these steps:

Step 1: Set the range of x values: 3 < x < 7

Step 2: Plug in each whole number value of x within the given range (4, 5, and 6) into the equation and calculate the corresponding y values.

For x = 4:
y = (4 - 5)^2 - 1
y = (-1)^2 - 1
y = 0

For x = 5:
y = (5 - 5)^2 - 1
y = (0)^2 - 1
y = -1

For x = 6:
y = (6 - 5)^2 - 1
y = (1)^2 - 1
y = 0

Step 3: Write the ordered pairs (x, y) based on the calculated y values.

For x = 4, the ordered pair is (4, 0)
For x = 5, the ordered pair is (5, -1)
For x = 6, the ordered pair is (6, 0)

In the given range, there are 3 ordered pairs (x, y) that satisfy the equation y=(x-5)^2-1.

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If Chris bought 5 tacos and 2 burritos for $13. 25 and  Brett bought 3 tacos and 2 burritos for $10. 75, the price of one taco is $1.25, and the price of one burrito is $3.50.

Let the price of one taco be T and the price of one burrito be B. We have the following equations:

5T + 2B = $13.25
3T + 2B = $10.75

To find the prices of the taco and the burrito, we can use the system of equations. First, subtract the second equation from the first equation:

(5T + 2B) - (3T + 2B) = $13.25 - $10.75
2T = $2.50

Now, divide by 2 to find the price of one taco:

T = $1.25

Next, plug the value of T back into one of the equations (let's use the second equation):

3($1.25) + 2B = $10.75
$3.75 + 2B = $10.75

Now, subtract $3.75 from both sides:

2B = $7.00

Finally, divide by 2 to find the price of one burrito:

B = $3.50

So, the price of one taco is $1.25, and the price of one burrito is $3.50.

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The base of her ladder should be 12 feet from the house.

A Pythagorean theorem is a useful theorem which can be applied so as to determine the length of the missing side of a right angled triangle. It states that:

/Hyp/^2 = /Adj/^2 + /Opp/^2

So that from the information given in the question, let the distance from the base of her ladder and the house be represented by x;

/Hyp/^2 = /Adj/^2 + /Opp/^2

20^2 = x ^2 + 16^2

400 = x^2 + 256

x^2 = 400 - 256

      = 144

x = 144^1/2

  = 12

x = 12 feet

Thus, Allison should place the base of her ladder 12 feet to the house.

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