The circumference of a circle is 15pi centimeters what is the area of the circle in terms of pi?

Answer:

Step-by-step explanation:

False.

The correct identity is:

cos^2(x/3) = 1/2[1+cos(2x/3)]

To see why, use the double angle formula for cosine:

cos(2x/3) = 2cos^2(x/3) - 1

Substitute this into the original equation:

cos(x/3)cos(x/3) = 1/2[1+2cos^2(x/3)-1]

Simplify:

cos^2(x/3) = 1/2[1+cos(2x/3)]

Answer:

Statement is true!

Step-by-step explanation:

Required to Prove:

[tex]\Large \textsf{$\cos \left(\frac{x}{3}\right)\cos \left(\frac{x}{3} \right)=\frac{1}{2} \left[1+\cos(\frac{2x}{3})\right]$}[/tex]

This is a special property, used in integral calculus, that can be derived and hence proved, from the double angle formula of cosine.

[tex]\large \textsf{Given that cos(A+B) = cosA\,cosB $-$ sinA\,sinB,}\\ \\\large \textsf{Hence cos(A+A) = cosA\,cosA $-$ sinA\,sinA}\\ \\\large \textsf{$\therefore$ cos2A = cos$^2$A $-$ sin$^2$A}\\ \large \textsf{$\rm \phantom{\therefore cos^2A}=$ 1 $-$ 2sin$^2$A}\\ \large \textsf{$\rm \phantom{\therefore cos^2A}=$ 2cos$^2$A $-$ 1 (using Pythagorean Identity $\Rightarrow cos^2A+sin^2A = 1$)}[/tex]

This property, can be quoted in exams and only has to be derived, not proved. Now using the Cos2A property, we can manipulate the formula:

[tex]\large \textsf{$\cos2\rm A = \cos^2A - \sin^2A$}\\ \\ \large \textsf{$\rm \phantom{\cos 2A}=2\cos^2A-1$}\\ \\ \large \textsf{$\rm \therefore \cos2A+1 = 2\cos^2A$}\\ \\ \large \textsf{$\rm \cos^2A=\frac{\cos2A+1}{2}$}\\ \\ \large \textsf{$\rm \phantom{\cos^2A}=\frac{1}{2}(\cos2A+1)$}\\ \\ \large \textsf{$\rm \phantom{\cos^2A}=\frac{1}{2}(1+\cos2A)$}[/tex]

And since:

[tex]\large \textsf{$\cos \left(\frac{x}{3}\right)\cos\left(\frac{x}{3}\right)=\cos^2\left(\frac{x}{3}\right)$}[/tex]

Therefore, inputting the value of A = [tex]\Large \textsf{$\frac{x}{3}$}[/tex] into the formula we derived above, hence:

[tex]\Large \boxed{\boxed{\textsf{$\cos \left(\frac{x}{3}\right)\cos \left(\frac{x}{3} \right)=\frac{1}{2} \left[1+\cos(\frac{2x}{3})\right]$}}} \Large \textsf{ , as required}[/tex]

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The limit of the trigonometric function f(x) = (1 - cos x) / x is equal to 0.

In this problem we need to determine the limit of a trigonometric function for x → 0. This can be done by simplifying the expression by trigonometric formulas. First, write the trigonometric function:

f(x) = (1 - cos x) / x

Second, modify the expression by means of algebra properties and trigonometric formulas:

f(x) = (2 / x) · (1 - cos x) / 2

f(x) = sin² (x / 2) / (x / 2)

f(x) = sin (x / 2) · [sin (x / 2) / (x / 2)]

For u = x / 2:

f(u) = sin u · (sin u / u)

Third, use limits to evaluate the trigonometric function:

f(u) = 0 · 1

f(u) = 0

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The probability that exactly 19 men and 7 women are chosen is 0.053107%.

Probability:

The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favorable outcomes and the total number of outcomes.

Probability of event to happen P(E) = Number of favorable outcomes/Total Number of outcomes.

[tex]C_n_,_x[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_n_,_x=\frac{n!}{x!(n-x)!}[/tex]

Desired outcomes:

19 men, from a set of 21

7 women, from a set of 11

[tex]D= C_2_1_,_1_9[/tex]  × [tex]C_1_1_,_7[/tex] [tex]=\frac{21!}{19!2!}[/tex] ×  [tex]\frac{11!}{7!4!}[/tex][tex]=69,300[/tex]

Total outcomes:

26 people from a set of 21 + 11 = 32.

[tex]T=C_3_2_,_2_6=\frac{32!}{26!6!}[/tex][tex]=13,049,164,800[/tex]

The probability is :

P = [tex]\frac{D}{T}= \frac{69,300}{13,049,164,800} = 5.3107[/tex]

0.053107% probability that exactly 19 men and 7 women are chosen.

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Answer:

Using the logarithmic identity log(a) + log(b) = log(ab), we can simplify the left-hand side of the equation:

log2(x-1) + log2(x+5) = log2((x-1)(x+5))

So the equation becomes:

log2((x-1)(x+5)) = 4

Using the exponential form of logarithms, we can rewrite the equation as:

2^4 = (x-1)(x+5)

Simplifying:

16 = x^2 + 4x - 5

Rearranging:

x^2 + 4x - 21 = 0

Using the quadratic formula:

x = (-4 ± sqrt(4^2 - 4(1)(-21))) / (2(1))

x = (-4 ± sqrt(100)) / 2

x = (-4 ± 10) / 2

So x = -7 or x = 3.

However, we need to check whether these solutions satisfy the original equation. We can see that x = -7 does not work, because both terms inside the logarithms would be negative. Therefore, the only solution is x = 3.

Answer:

Using the properties of logarithms, we can simplify the left-hand side of the equation:

log2(x-1) + log2(x+5) = log2((x-1)(x+5))

Therefore, the equation becomes:

log2((x-1)(x+5)) = 4

Using the definition of logarithms, we can rewrite this equation as:

2^4 = (x-1)(x+5)

16 = x^2 + 4x - 5

Simplifying further:

x^2 + 4x - 21 = 0

We can now use the quadratic formula to solve for x:

x = (-4 ± sqrt(4^2 - 4(1)(-21))) / (2*1)

x = (-4 ± sqrt(100)) / 2

x = (-4 ± 10) / 2

x = -7 or x = 3

However, we need to check if these solutions satisfy the original equation.

When x = -7:

log2(x-1) + log2(x+5) = log2((-7-1)(-7+5)) = log2(16) = 4

So x = -7 is a valid solution.

When x = 3:

log2(x-1) + log2(x+5) = log2((3-1)(3+5)) = log2(16) = 4

So x = 3 is also a valid solution.

Therefore, the solutions to the equation log2(x-1) + log2(x+5) = 4 are x = -7 and x = 3.

Step-by-step explanation:

The answer choice that matches the calculated confidence interval is 62.9 to 83.1

In statistics, the mean is a measure of central tendency that represents the average value of a set of numerical data. It is also known as the arithmetic mean, and it is calculated by adding up all the values in the dataset and dividing the sum by the number of values.

The mean is a useful measure of central tendency because it is easy to calculate, and it provides a single value that represents the center of the dataset. It is affected by outliers, which are extreme values that are far from the other values in the dataset, so it may not accurately represent the typical value of the data if there are outliers present.

To construct a 95.44% confidence interval for the mean score, u, of all students taking the test, we can use the formula:

CI = x ± t(alpha/2, n-1) * (s / √(n))

where CI is the confidence interval, x is the sample mean (73), t(alpha/2, n-1) is the t-value for the given alpha level (0.0278) and degrees of freedom (n-1=4) from the t-distribution table, s is the sample standard deviation, and n is the sample size.

The sample standard deviation is not given, so we will assume that it is the same as the population standard deviation, which is 15. Thus, s = 15.

Using the t-distribution table with 4 degrees of freedom and an alpha level of 0.0278, we find that the t-value is approximately 3.747.

Plugging in the values into the formula, we get:

CI = 73 ± 3.747 * (15 / √(5))

Simplifying, we get:

CI = 73 ± 16.27

Therefore, the 95.44% confidence interval for the mean score, u, of all students taking the test is:

CI = (73 - 16.27, 73 + 16.27)

CI = (56.73, 89.27)

Rounding to one decimal place, we get:

CI = (56.7, 89.3)

Therefore, the answer choice that matches the calculated confidence interval is:

62.9 to 83.1

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The complete question is:

The mean score, x on an aptitude test for a random sample of 5 students was 73, assuming that 0 = 15, construct a 95.44% confidence interval for the mean score, u of all students taking the test. answer choices, 43 to 103, 59.6 to 86.4, 62.9 to 83.1, and 67.0 to 79.0.

1). There are 5,245,786 different number combinations that could win. The likelihood of taking home the lottery's grand prize is 1 in 5,245,786 or roughly 0.000019%.

2). There are 42,467,328,000 different winning number combinations that a gambler can choose from.

Combinations are the various ways, independent of their sequence, in which a group of things or objects can be chosen.

The formula n! / (r! * (n-r)! can be used to determine the number of potential combinations of r items from a collection of n items, which is symbolised by the symbol C(n,r).

1. Six numbers are chosen at random from a pool of 42 numbers in a 6/42 lottery. The formula for combinations can be used to determine how many winning number combinations a gambler has the option of choosing:

C(42, 6) = 42! / (6! * (42-6)!)

= 5,245,786

2. The number of winning number combinations that a bettor may choose to select can be determined using the formula for permutations with repetition if the 6/42 Lottery permits repeat of numbers:

[tex]42^6[/tex] = 42 * 42 * 42 * 42 * 42 * 42 = 42,467,328,000

There are therefore 42,467,328,000 different ways to pick winning numbers. The odds of taking home the lottery's grand prize are 1 in 42,467,328,000, or roughly 0.000000002%.

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Answer:

The correct equation to find the total number of bags of glass and plastic Silas will take to the recycling center is C. x - 18 = 6.

In this equation, x represents the total number of bags of glass and plastic Silas will take to the recycling center. The left side of the equation represents the number of bags of glass Silas will take to the recycling center (x - 18), and the right side represents the number of bags of plastic he still needs to take to the recycling center (6).

By setting the two expressions equal to each other (x - 18 = 6), we can solve for x and determine the total number of bags Silas will take to the recycling center. Adding 18 to both sides of the equation gives us x = 24, which means Silas will take a total of 24 bags (18 bags of glass and 6 bags of plastic) to the recycling center.

Area of the two right triangles:

A = 1/2(b)(h)

A = 1/2(10)(24)

A = 120

Total area = 240

Area of the left-most rectangle:

A = (b)(h)

A = (24)(25)

A = 600

Area of the right-most rectangle:

A = (b)(h)

A = (25)(26)

A = 650

Area of the base rectangle:

A = (b)(h)

A = (10)(25)

A = 250

Surface Area:

240 + 600 + 650 + 250

1740

Answer: 1740 cm^2

Hope this helps!

[tex]\sf SA=\boxed{\sf 1740cm^{2} }.[/tex]

Check attached 1 to see what parts we're referring to in this step.

This part forms a right triangle. Therefore, the formula to use to find it's area is the following:

[tex]\sf A=\dfrac{bh}{2}[/tex]; where "b" is the length of the base of the triangle, and "h" is its height.

Since we have another section identical to this part at the back, we multiply this area by 2 and calculate:

[tex]\sf A=2\dfrac{bh}{2}=(10cm)(24cm)=240cm^{2}[/tex]

Check image 2 to see this part highlighted.

This shape forms a rectangle. Therefore, use the following formula to calculate:

[tex]\sf A=lw[/tex]; where "l" is length, and "w" is width.

[tex]\sf A=(25cm)(10cm)=250cm^{2}[/tex]

Check image 3.

This shape also forms a rectangle, therefore its area is calculated like this:

[tex]\sf A=(24cm)(25cm)=600cm^{2}[/tex]

Check image 4.

This shape also forms a rectangle, therefore its area is calculated like this:

[tex]\sf A=(26cm)(25cm)=650cm^{2}[/tex]

The total surface area of this prism is given by the addition of all of its individual areas that we just calculated.

[tex]\sf SA=240cm^{2} +250cm^{2} +600cm^{2} +650cm^{2} =\boxed{\sf 1740cm^{2} }.[/tex]

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The missing sides of each geometric system are summarized below:

Case 9: (x, y) = (9, 12)

Case 12: (x, y) = (√51, 7)

Case 15: (x, y) = (8, 15)

In this question we find three cases of geometric systems formed by two right triangles, all missing sides can be found by means of Pythagorean theorem:

r = √(x² + y²)

Where:

Now we proceed to determine all missing sides:

Case 9

y = √(15² - 9²)

y = 12

x = 9

Case 12

x = √(10² - 7²)

x = √51

y = 7

Case 15

y = √(17² - 15²)

y = 8

x = 15

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A) (2x+1)⁰ = 96⁰

=> 2x = 96⁰ - 1⁰ = 95⁰

=> x = 95⁰/2 = 47⁰30' (= 47.5⁰)

B) x⁰ = (2x-7)⁰

=> x - 2x = -7

=> -x = -7

=> x = 7⁰

C) mIJ = 45⁰ ; mJK = 57⁰

m✓ IJK = 180⁰- 45⁰ - 57⁰ = 78⁰

=> mIK = 78⁰

Ans: a) 47.5⁰ b) 7⁰ c) mIJ = 45⁰ ; mJK = 57⁰ ; mIK = 78⁰

Ok done. Thank to me >:333

Answer:

4. n + 24

5. b - 11

6. d/5

The real distance between City X and City Y is 17 miles.

We know that the scale of the drawing is:

1 inch = 17 miles.

Now, if you look at the diagram for cities X and Y, you can see that the distance between City X and City Y is exactly 1 inch.

And we know that 1 inch is equivalent to 17 miles, then we can conclude that the actual distance between the two cities is exactly 17 miles.

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The missing angle C is 103, length of a is 9 m, and length b is 16 m.

option B.

The missing angle C is calculated as follows;

A + B + C = 180 (sum of angles in a triangle)

26 + 51 + C = 180

C = 180 - 77

C = 103

The value of length a and length b is calculated as follows;

sin 26/a = sin 103/20

0.438/a = 0.0487

a = 0.438/0.0487

a = 9 m

b/sin51 = 20/sin103

b = 16 m

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Step-by-step explanation:

(a) The lower limit of the 90% confidence interval can be calculated using the formula:

lower limit = sample mean - margin of error

Plugging in the given values, we have:

lower limit = 7.43 - 1.32

lower limit = 6.11 (rounded to 2 decimal places)

Therefore, the lower limit of the 90% confidence interval is approximately 6.11 hours per night.

(b) If 500 random samples like this were selected, and a 90% confidence interval was constructed using each sample, the expected number of intervals that would contain the population mean can be approximated using the margin of error as a guide.

Since the margin of error is 1.32 hours, we can expect roughly 90% of the confidence intervals to contain the true population mean. Therefore, out of 500 samples, we would expect approximately:

500 * 0.9 = 450

So, approximately 450 of these confidence intervals would contain the population mean.

The value of x, based on the Alternate Interior Angles Theorem, is calculated as: x = 5.

The Alternate Interior Angles Theorem states that if two parallel lines are intersected by a transversal, then the pairs of alternate interior angles formed are congruent. In other words, if two lines are parallel and a third line intersects them, then the angles that are inside (or "interior" to) the two parallel lines and on opposite sides of the transversal are congruent.

Therefore, we have:

3x - 8 = x + 2 [based on the Alternate Interior Angles Theorem]

Combine like terms:

3x - x = 8 + 2

2x = 10

2x/2 = 10/2

x = 5

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The equation which represents the rule for the variable values is given by option c. y = 2x + 2

The values in the table are,

x  -2   -1    0   1   2

y   -2    0   2   4  6

let us consider two coordinates of the given values in the table .

( x₁ , y₁ ) = ( -2 , -2 )

( x₂ , y₂ ) = ( 0 , 2 )

Using the formula for the slope intercept form of the line we get the equation,

( y - y₁ ) / ( x - x₁ ) = ( y₂ - y₁ ) / ( x₂ - x₁ )

Substitute the values to get the equation of the line we have,

⇒ ( y - ( - 2 ) ) / ( x -  ( -2 ) ) = ( 2 - ( - 2 ) ) / ( 0 - ( - 2 ) )

⇒  ( y + 2 ) / ( x + 2 ) = ( 2 + 2 ) / ( 0 + 2)

⇒  ( y + 2 ) / ( x + 2 ) = 4  / 2

⇒  ( y + 2 ) / ( x + 2 ) = 2

⇒ y + 2 = 2 ( x + 2)

⇒ y + 2 = 2x + 4

⇒ y = 2x + 4 - 2

⇒ y = 2x + 2

Therefore, the equation which represents the rule for the given values of the variable is equal to option c. y = 2x + 2

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true

Step-by-step explanation:

all square have the same features and properties like

all side are equal

The function that models the population t years after 2000 is P(t) = 14400 * (1.04)^t

The population growth function is of the form:

P(t) = P₀ * (1 + r)^t

Where:

P(t) is the current population after t years

P₀ is the starting population

r is the annual growth rate in percent

Thus, P₀ = 14400 and 4% = 0.04

P(t) = P₀ * (1 + r)^t

P(t) = 14400 * (1 + 0.04)^t

P(t) = 14400 * (1.04)^t

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Complete Question

The fox population in a certain region has an annual growth rate of 4 percent per year. It is estimated that the population in the year 2000 was 14400.

a) Find a function that models the population t years after 2000 (t=0 for 2000).

Your answer is P(t) =

The probability of selecting an R or an E without replacement, and then selecting an S is 5/36

Because the word SEPTEMBER has 9 letters, there are 9 different alternatives for the initial letter.

The probability of selecting a R or an E without replacing is 2+3=5.

The odds of picking a R or an E on the initial draw are 5/9.

After the first letter is drawn, the bag contains eight letters, including one S. If the first letter is not replaced, there are only four letters that fit the requirement.

Given that a R or an E was selected without replacement on the first draw, the probability of selecting a S on the second draw is 4/8.

When we multiply these probability together, we get:

P(R or E, not replacing) * P(S after R or E, not replacing) = (5/9) * (4/8) = 10/72 = 5/36

Hence, the probability of selecting an R or an E without replacement, and then selecting an S is 5/36

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The answers are explained in the solution.

Considering the triangle, ABC,

BC = √AC²-AB² [Pythagoras theorem]

BC = √126.4²-126²

BC = 10 ⇒ Height of the ramp at B,

Slope = tanBC/AB = 10/126

The slope is less than 1/12, hence, it will get ADA approval,

Let θ be angle of elevation,

θ = tan⁻¹(10/126)

= 4.5° < 5°

Hence the town met the given requirement.

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Answer: ∠DGB or ∠EGA

Step-by-step explanation:

Supplementary: Either of two angles whose sum is 180°.

Starting Angle: ∠DGE

Possible Supplements: ∠DGB or ∠EGA

I hope this helps ^^

Answer:

I hope this helps...

Please mark me brainliest

Answer:

Step-by-step explanation:

1 out of 100 is 1 percent. This is because percentage is always out of 100 so you don’t have to change anything. That means 1 is always 1 percent of 100. It is also 1 part of 100.

The equations solved for the variables T₁ and P₁ are:

We start with the equation:

(P₁*V₁)/T₁ = (P₂*V₂)/T₂

And we want to solve this for T₁, we can multiply both sides by T₁ and divide both sides by the expression in the right side.

(P₁*V₁) = T₁*[ (P₂*V₂)/T₂]

(P₁*V₁)*[T₂/(P₂*V₂)] = T₁

That is the equation solved for T₁.

34: Now we have the same equation but we want to solve it for P₁, to do so, just multiply both sides by T₁/V₁

We will get:

(T₁/V₁)*(P₁*V₁)/T₁= (T₁/V₁)*(P₂*V₂)/T₂

P₁ = (T₁/V₁)*(P₂*V₂)/T₂

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The probability of rolling a number less than 9 is 1

The probability of rolling a number less than 9 is 1, since all the possible outcomes are 1, 2, 3, 4, 5, 6 and all of them are less than 9.

The set of equally likely outcomes when rolling a 6-sided die is {1, 2, 3, 4, 5, 6}.

There are no outcomes greater than 6 since that is the maximum number on the die.

Therefore, the probability of rolling a number less than 9 is equal to the probability of rolling any number on the die, which is 1, since all outcomes in the sample space are equally likely.

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Answer:

The image you provided appears to be a rectangular prism. To find the surface area of a rectangular prism, we need to add up the areas of all of its faces.

The rectangular prism has dimensions of 4 cm x 6 cm x 8 cm.

Each face of the rectangular prism is a rectangle, so the area of each face can be found by multiplying the length by the width.

The surface area of the rectangular prism is:

2(4 cm x 6 cm) + 2(4 cm x 8 cm) + 2(6 cm x 8 cm)

= 48 cm^2 + 64 cm^2 + 96 cm^2

= 208 cm^2

Therefore, the surface area of the rectangular prism is 208 square centimeters.

have a good day and stay safe

the answer is the fourth option

a < -4 1/2

The Amplitude is 1/2 and period is π/2.

We have the function as

y= 1/2 cos π/4 x

As, The general equation of a Cosine function is

y=A cos (B(x−D))+C

where A is Amplitude , D is the shift.

So, the amplitude is 1/2

Period = 2π / 4= π/2

and, the phase shift is not possible to determine.

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