Prove that
sin 2x
1+ cos2x
= tan x

IQR offers a more accurate indicator of consistency than range since it is less impacted by outliers.

Due to Bus 18's smaller IQR, we can infer that it is more consistent than Bus 47.

An indicator of statistical dispersion used to describe the variability of a dataset is the interquartile range (IQR). It is derived by subtracting a dataset's 75th percentile (Q3) from its 25th percentile (Q1). When compared to other measures of variability like range, the IQR represents the middle 50% of the data and is less susceptible to outliers.

We must contrast the measurements of variability for the two buses in order to determine which one is more reliable. We can utilize range and interquartile range as measurements of variability. (IQR).

The range of a dataset is the difference between its maximum and minimum values.

The range for Bus 18 is 16 (22 - 6), whereas the range for Bus 47 is 24. (28 - 4).

A measure of variability that is less impacted by outliers than the range is the interquartile range (IQR). It is determined as the difference between a dataset's third and first quartiles (Q3 and Q1, respectively).

The IQR is 16 for Bus 18 (Q3 - Q1 = 22 - 6)

IQR offers a more accurate indicator of consistency than range since it is less impacted by outliers. Due to Bus 18's smaller IQR, we can infer that it is more consistent than Bus 47.

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Answer:

40 units

Step-by-step explanation:

a = 8

b = 15

To find c, we can use the formula: [tex]a^{2} +b^{2} =c^{2}[/tex]

[tex]8^{2} +15^{2} =x^{2}[/tex]

64 + 225 = c^2

289 = c^2

c = 17

a + b + c = perimeter

8 + 15 + 17 = 40

Answer:

  40 units

Step-by-step explanation:

You want the perimeter of the triangle shown in the graph.

You can count the grid squares to find the horizontal and vertical dimensions of the triangle. You find they are 8 units and 15 units, respectively.

The length of the slant side is the hypotenuse of a right triangle with sides 8 and 15. If you don't recognize this {8, 15, 17} Pythagorean triple, you can find the hypotenuse using the Pythagorean theorem:

  c² = a² +b²

  c² = 8² +15² = 64 +225 = 289

  c = √289 = 17

The long side of the triangle is 17 units.

The perimeter of the triangle is the sum of the lengths of its sides:

  P = 8 + 15 + 17 = 40

The perimeter is 40 units.

__

Additional comment

A "Pythagorean triple" is a set of three integer side lengths that form a right triangle. The triple is "primitive" if the numbers have no common factor. There are a few Pythagorean triples that regularly show up in algebra, trig, and geometry problems. Some of them are ...

  {3, 4, 5}, {5, 12, 13}, {7, 24, 25}, {8, 15, 17}, {9, 40, 41}

You will also see multiples of these, for example, 2·{3, 4, 5} = {6, 8, 10}.

The smallest is {3, 4, 5}, and it is the only set that is an arithmetic sequence (has constant differences between lengths). In every case, the sum of the numbers is even. (A right triangle cannot have integer side lengths and an odd perimeter value.)

The answer of the given question based on the graph is ,  The locations of A′ and B′ are A′ (0, 1) and B′ (1, 0); lines f and f′ are parallel.

A scale factor is a number that scales, or multiplies, a quantity by some factor. It is used in mathematics to describe the relationship between corresponding measurements of two similar figures, such as triangles or rectangles.

To dilate line f by scale factor of one half with center of dilation at origin, we  multiply coordinates of each point on line f by 1/2.

The equation of line f can be found by using the points A and B:

slope of line f = (0 - 2)/(2 - 0) = -1

y-intercept of line f = 2

Therefore, the equation of line f is y = -x + 2.

To find the coordinates of A' and B' after dilation, we can apply the dilation factor to each point:

A' = (0, 2)*1/2 =(0, 1)

B' = (2, 0)*1/2 =(1, 0)

So A' is located at (0, 1) and B' is located at (1, 0) after dilation.

Now let's analyze the relationship between lines f and f'. The dilation was centered at the origin, so the origin is a fixed point of the dilation. This means that the point where lines f and f' intersect must be the origin.

If we plug in x = 0 into the equation of line f, we get y = 2. This means that point A is located at (0, 2) and intersects with line f at y = 2. After dilation, point A' is located at (0, 1), which means that lines f and f' intersect at point A.

To determine the relationship between lines f and f', we can compare their equations. The equation of f' can be found by using the points A' and B':

slope of f' = (0 - 1)/(1 - 0) = -1

y-intercept of f' = 0

Therefore, the equation of f' is y = -x.

Comparing the equations of f and f', we can see that they have the same slope of -1, which means they are parallel. Therefore, the correct answer is: The locations of A′ and B′ are A′ (0, 1) and B′ (1, 0); lines f and f′ are parallel.

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Answer:

x = -3 and x = 1 are the two zeroes of this function. Choose one of them for your answer.

Answer:

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

Use the formula for permutations:

The number of objects is n = 7 books.

Calculate the factorial:

Answer:1111

Step-by-step explanation:

a) The angular velocity of the wheel is approximately 1.0667 radians per second. b) The wheel will spin at approximately 10.16 revolutions per minute.

Angular velocity is a measure of how quickly an object rotates or revolves around a fixed point, such as an axis or a center of rotation. It is a vector quantity that describes the rate of change of an object's angular position with respect to time, and is usually expressed in units of radians per second (rad/s) or degrees per second (deg/s).

a) To find the angular velocity of the wheel, we can use the formula:

angular velocity = linear velocity / radius

In this case, the linear velocity of the hamster is 16 centimeters per second, and the radius of the wheel is 15 centimeters. So, we have:

angular velocity = 16 / 15

angular velocity = 1.0667 radians per second

Therefore, the angular velocity of the wheel is approximately 1.0667 radians per second.

b) To find the speed of the wheel in revolutions per minute, we need to convert the angular velocity from radians per second to revolutions per minute. There are 2π radians in one revolution, and 60 seconds in one minute. So, we can use the following formula:

angular velocity (in revolutions per minute) = angular velocity (in radians per second) * 60 / 2π

Plugging in the angular velocity we found in part (a), we get:

angular velocity (in revolutions per minute) = 1.0667 * 60 / 2π

angular velocity (in revolutions per minute) ≈ 10.16 revolutions per minute

Therefore, the wheel will spin at approximately 10.16 revolutions per minute.

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Answer:

If ticket sales were $3,000 for 400 tickets then 280 students attended the play.

Step-by-step explanation:

tan = 1/√3= perpendicular/ base

1/√3 = AB / AC

AB / 6 = 1/√3

AB = 6/√3

The histogram should be modified as follows:

A histogram is a type of graphical representation that displays the distribution of a dataset. It is a bar graph-like chart where the data is divided into intervals, called "bins", which are represented by adjacent rectangular bars of varying heights.

Then the height of the histogram gives the number of observations into each data interval.

Hence the histogram should be modified as follows:

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The probability of drawing a vowel, placing it back in the bag, and then drawing a consonant is 0.1875 or 18.75%.

The probability of drawing a vowel from the bag of random letter tiles is 25%. Since the tile is replaced after drawing, the probability of drawing a vowel on the second draw is also 25%.

The probability of drawing a vowel and then a consonant can be calculated by multiplying the probabilities of each event:

P(vowel and consonant) = P(vowel) x P(consonant)

= 0.25 x 0.75

= 0.1875

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The experimental probability of spinning a 4 will equals to 0.3.

Outcome   Tally   Frequency

1                   IIII          4

2                  III           3

3                  IIIIIIII       7

4                  IIIIII        6  

                   Total     20

The probability of spinning a 4:

= 6/20

= 3/10

= 0.3

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Answer: The formula for compound interest is:

A = P(1 + R/100)^t

where A is the amount after t years, P is the principal amount, R is the rate of interest per annum, and t is the time period in years.

Here, P = Rs. 3,500, R = 8%, and t = 2 years.

So, the amount after 2 years will be:

A = 3,500(1 + 8/100)^2

= 3,500(1.08)^2

= 3,892.32

Therefore, the compound interest for 2 years will be:

CI = A - P

= 3,892.32 - 3,500

= 392.32

Hence, the compound interest on Rs. 3,500 for 2 years at the rate of 8% per annum is Rs. 392.32.

Step-by-step explanation:

The missing step in the proof is: D. ADCA

In the statements, we are given that AD || BC and ZDAC = ZBCA (alternate interior angles theorem), so we can conclude that triangle ADCA is similar to triangle BCA (by AA similarity criterion).

Therefore, we have the following proportional sides:

DC/AC = AC/BC

Given that DC = BA (given statement), we can substitute BA for DC in the proportion:

BA/AC = AC/BC

Now, we can cross-multiply to get:

BA * BC = AC²

Using the given statement that AC = AC (reflexive property of congruence), we can substitute AC for AC in the equation:

BA * BC = AC * AC

Use the definition of congruent sides to replace BA with DC:

DC * BC = AC * AC

And finally, use the substitution property of equality to replace BC with 6 units (given statement):

DC * 6 = AC * AC

From this equation, we can see that DC = 6 units, which completes the proof. Therefore, the missing step is statement D. ADCA.

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The value of GH in the right angle triangle, given FH and Angle FHG is 22.46 inches.

Since we have the adjacent side (FH) and want to find the hypotenuse (GH), we can use the cosine function.

cos(35°) = adjacent side (FH) / hypotenuse (GH)

We know that FH = 18.4 in. So, we can write the equation as:

cos(35°) = 18.4 / GH

GH = 18.4 / cos(35°)

GH = 18.4 / 0.81915

=  22.46 inches

So, the length of GH, the hypotenuse, is approximately 22.46 inches.

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Answer:

C

Step-by-step explanation:

using the sine ratio and the exact value

sin45° = [tex]\frac{1}{\sqrt{2} }[/tex] , then

sin45° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{5}{x}[/tex] = [tex]\frac{1}{\sqrt{2} }[/tex] ( cross- multiply )

x = 5[tex]\sqrt{2}[/tex]

Answer: The area of the door can be calculated as:

Area of door = width x height

= 0.9 m x 2.1 m

= 1.89 square meters

The area of one window can be calculated as:

Area of window = width x height

= 1.5 m x 1.2 m

= 1.8 square meters

Since there are four windows, the total area of the four windows is:

Total area of four windows = 4 x Area of window

= 4 x 1.8 square meters

= 7.2 square meters

Therefore, the total area of the door and the four windows is:

Total area = Area of door + Total area of four windows

= 1.89 square meters + 7.2 square meters

= 9.09 square meters

Hence, the total area of the door and the four windows is 9.09 square meters.

Step-by-step explanation:

The smaller number is 6 .

The larger number is 22

Let's let x be the smaller number and y be the larger number.

From the problem, we know that one number is eight less than five times the other, so we can write:

y = 5x - 8

We also know that the sum of the two numbers is 28, so we can write:

x + y = 28

Now we have two equations in two variables. We can solve for one of the variables in terms of the other, and substitute that expression into the other equation to eliminate one variable.

Let's solve the first equation for x

x = (y + 8)/5

Now we can substitute this expression for x into the second equation:

(y + 8)/5 + y = 28

Multiplying both sides by 5 to eliminate the fraction, we get:

y + 8 + 5y = 140

Combining like terms, we get:

6y + 8 = 140

Subtracting 8 from both sides, we get:

6y = 132

Dividing both sides by 6, we get:

y = 22

Now we can use the equation y = 5x - 8 to solve for x:

22 = 5x - 8

Adding 8 to both sides, we get:

30 = 5x

Dividing both sides by 5, we get:

x = 6

Therefore, the smaller number is 6 and the larger number is 22.

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Answer:

hm

Step-by-step explanation:

The reason why it is possible to draw two different rectangles that have an area of 30 square inches is that there are multiple pairs of whole numbers that multiply to 30. For example, a rectangle with dimensions 5 inches by 6 inches has an area of 30 square inches, and so does a rectangle with dimensions 2 inches by 15 inches.

However, the number 7 is a prime number, which means it can only be divided evenly by 1 and itself. Therefore, the only way to get an area of 7 square inches for a rectangle is by multiplying 1 and 7, which means the dimensions of the rectangle would be 1 inch by 7 inches. Since there are no other pairs of whole numbers that multiply to 7, it is not possible to draw two different rectangles with an area of 7 square inches.

The correct answer is: A. Y intercept will be less and X will be less.

A linear function's slope is a gauge of how steep the line is. Between any two points on the line, it is the ratio of the change in the y-coordinate to the change in the x-coordinate. The formula is used to compute it:

slope equals (y2 - y1)/. (x2 - x1)

where any two points on the line (x1, y1) and (x2, y2) are.

The company's expenses will rise once it begins paying a driver at a rate of 0.25 cents per mile, which means its profit will decline. The profit curve on the y-axis will go vertically downward as a result of this. But, because the cost of paying the driver is fixed per mile and has no impact on the profit per mile, the slope of the line—which reflects the profit per mile—will stay the same.

Hence, A. Y intercept will be less and X will be less is correct.

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Answer:

-2.5

Step-by-step explanation:

follow BIMDAS.

like my answer if you find it helpful

Answer:

Step-by-step explanation:

Answer:

You can make 12 possible sundaes with these toppings.

Step-by-step explanation:

Chocolate, Vanilla, and Strawberry all have 4 possible outcomes:

1. Fudge & Chocolate Sprinkles

2. Fudge & Rainbow Sprinkles

3. Marshmallow & Chocolate Sprinkles

4. Marshmallow & Rainbow Sprinkles

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

4 x 3 will equal 12, the total possible sundaes you can make with these toppings and ice cream flavors.

Answer: Since we know that cos A = 3/10, we can use the Pythagorean identity to find sin A:

sin^2 A + cos^2 A = 1

sin^2 A = 1 - cos^2 A

sin A = sqrt(1 - cos^2 A)

Substituting cos A = 3/10, we get:

sin A = sqrt(1 - (3/10)^2)

sin A = sqrt(1 - 9/100)

sin A = sqrt(91)/10

Since we also know that tan A < 0, we know that the sine and tangent of A have opposite signs, and therefore sin A is negative.

Therefore:

sin A = -sqrt(91)/10

Step-by-step explanation:

Answer:

Step-by-step explanation:

We get triangle PQR by plotting the point P (1, 4), Q (3, 1), R (2, -1) on the graph paper when rotated through 180° about the origin. The new position of the point is: P (1, 4) → P' (-1, -4) Q (3, 1) → Q' (-3, -1) R (2, -1) → R' (-2, 1) Thus, the new position of ∆PQR is ∆P’Q’R’.

Answer:

B.

Step-by-step explanation:

Answer:

B.

Step-by-step explanation:

the description is very confusing and lacks any hint about the behavior of the curve before the change.

but ok, let's assume that the curve (line ?) is in general going up. in other words, if the traveled miles go up, so does the profit.

I also don't know how they could drive the miles without paying the driver, but again, let's assume they did.

now they pay the driver. $0.25 per mile.

this has to come out of their profit.

so, the Y (profit) values all go down by 0.25.

therefore, the y-intercept is going down too.

that eliminates C. and D.

for this to be true they have to have some overhead costs that apply even if there are 0 miles traveled (which is the meaning of the y-intercept : the y-value when x = 0).

so, the line must be in a form

y = ax + b

with "b" <> 0.

and since we assumed the line to go up (positive slope), a "sinking" line means that the x-intercept (the break-even point, where the line goes from below the x-axis and therefore negative (y) profit up into positive profit) will move to the right (become greater).

because now that they have additional costs (paying the driver), it takes longer (more driven miles) to make a positive profit.

and that eliminates A, leaving B. as the right answer.

Answer:

Least: 565

Greatest: 574

Step-by-step explanation:

564 would round down to 560 and 575 would round to 580

Helping in the name of Jeus.

The least whole number that rounds to 570 when rounded to the nearest ten is 570. The greatest whole number that rounds to 570 when rounded to the nearest ten is 580.

To find the least whole number that rounds to 570 when rounded to the nearest ten, we need to look at the digit in the tens place. In this case, the tens digit is 7. Since the number can only round up or down, we need to determine if rounding 570 to the nearest ten would be up or down. Since 570 is closer to 570 than to 580, we can round down to 570. Therefore, the least whole number that rounds to 570 when rounded to the nearest ten is 570 itself.

On the other hand, to find the greatest whole number that rounds to 570 when rounded to the nearest ten, we again need to look at the digit in the tens place. Since the tens digit is 7, we need to determine if rounding 570 to the nearest ten would be up or down. Since 570 is closer to 580 than to 570, we can round up to 580. Therefore, the greatest whole number that rounds to 570 when rounded to the nearest ten is 580.

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The probability of choosing 5 CDs in alphabetical order out of a collection of 23 CDs is approximately 0.00003 or 0.003%.

What is combination?

In mathematics, a combination is a way of selecting objects from a larger group of objects, where the order of selection is not important. A combination is a subset of objects that are chosen from a larger set without regard to the order in which they are selected.

The formula  is given by:

C(n, r) = [tex]\frac{n!}{r!* (n - r)!}[/tex]

The total number of ways to choose 5 CDs from a collection of 23 CDs is given by the combination formula:

C(23, 5) = [tex]\frac{23!}{5!* (23 - 5)!}[/tex]

             = 33649

This means there are 33649 different ways to arrange 5 CDs out of the 23 CDs.

Out of these 33649 ways, only one arrangement is alphabetical. For this to happen, the first CD chosen must be the first one alphabetically, the second CD chosen must be the second one alphabetically, and so on.

The first CD can be any of the 23 CDs. However, once the first CD is chosen, there is only one CD that can be the second one alphabetically, two CDs that can be the third one alphabetically, and so on. Therefore, the probability of choosing 5 CDs in alphabetical order is:

P = 1 / C(23, 5)

P = 1 / 33649

P ≈ 0.00003

So the probability of choosing 5 CDs in alphabetical order out of a collection of 23 CDs is approximately 0.00003 or 0.003%.

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