A whole theater has the capacity to hold 1,000 people. On the opening night, the theater was 95% full. What part of the theater was filled on opening night?

The answer to the trigonometry question in the picture attached is:

   = cos θ / [sin θ * (1 - sin θ)] * (1 + sin θ)

Simplify 1-csc θ as follows:

1 - csc θ = (1 - csc θ)(1 + csc θ) / (1 + csc θ)

= 1 - csc^2 θ / (1 + csc θ)

= 1 - 1/sin^2 θ / (1 + 1/sin θ)

= 1 - sin^2 θ / (sin θ + 1)

= (sin θ - sin^2 θ) / (sin θ + 1)

Simplify 1+csc θ as follows:

1 + csc θ = (1 + csc θ)(1 - csc θ) / (1 - csc θ)

= 1 - csc^2 θ / (1 - csc θ)

= 1 - 1/sin^2 θ / (1 - 1/sin θ)

= 1 - sin^2 θ / (sin θ - 1)

= (sin θ + sin^2 θ) / (sin θ - 1)

Substitute the above simplifications in the expression cos θ/(1-csc θ) * 1+csc θ/(1+ csc θ) to get:

cos θ / (sin θ - sin^2 θ) * (sin θ + sin^2 θ) / (sin θ + 1)

Simplify the expression by canceling out the sin^2 θ terms:

cos θ / (sin θ - sin^2 θ) * (sin θ + sin^2 θ) / (sin θ + 1)

= cos θ / (sin θ - sin^2 θ) * (1 + sin θ) / (sin θ + 1)

Simplify further by factoring out common terms in the numerator and denominator:

cos θ / (sin θ - sin^2 θ) * (1 + sin θ) / (sin θ + 1)

= cos θ * (1 + sin θ) / [(sin θ - sin^2 θ) * (sin θ + 1)]

Finally, simplify the expression by factoring out a sin θ term from the denominator:

cos θ * (1 + sin θ) / [(sin θ - sin^2 θ) * (sin θ + 1)]

= cos θ * (1 + sin θ) / [sin θ * (1 - sin θ) * (sin θ + 1)]

= cos θ / [sin θ * (1 - sin θ)] * (1 + sin θ)

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Answer:

3

Step-by-step explanation:

f(x)=7xe^x,a=0,7x + 7x^2 + (35/3)x^3 + (7/2)x^4.

The function f(x) is given by:

f(x) = 7xe^xTo

find the first four nonzero terms of the series for f(x) centered at a=0 using the definition of a Taylor series, we need to use the formula:

f(a) + f'(a)(x-a)/1! + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + f''''(a)(x-a)^4/4! + ...

The first four nonzero terms will be obtained when we compute the first four derivatives of f(x) and evaluate them at x=a=0.

f(a) = f(0) = 7(0)e^0 = 0

f'(x) = 7e^x + 7xe^x = 7e^x(1+x)

f'(a) = f'(0) = 7e^0(1+0) = 7

f''(x) = 7e^x + 7e^x + 7xe^x = 14e^x + 7xe^x

f''(a) = f''(0) = 14e^0 + 7(0)e^0 = 14

f'''(x) = 14e^x + 14e^x + 7e^x + 7xe^x = 28e^x + 7e^x(1+x)

f'''(a) = f'''(0) = 28e^0 + 7e^0(1+0) = 35

f''''(x) = 28e^x + 28e^x + 14e^x + 7xe^x = 70e^x + 14e^x(1+x)

f''''(a) = f''''(0) = 70e^0 + 14e^0(1+0) = 84

Thus, the first four nonzero terms of the series for f(x) centered at a=0 are:

f(x) = f(a) + f'(a)(x-a)/1! + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3!

= 0 + 7(x-0)/1! + 14(x-0)^2/2! + 35(x-0)^3/3! + 84(x-0)^4/4!

= 7x + 7x^2 + (35/3)x^3 + (7/2)x^4

Hence,

f(x)=7xe^x, a=0, the first four nonzero terms of the series are 7x + 7x^2 + (35/3)x^3 + (7/2)x^4.

f(x)=7xe^x,a=0,7x + 7x^2 + (35/3)x^3 + (7/2)x^4.

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a) The 90% confidence interval for the population proportion is approximately (0.7777, 0.8423).

b) The 95% confidence interval for the population proportion is approximately (0.7737, 0.8463).

To calculate the confidence intervals for the population proportion, we can use the formula:

Confidence Interval = sample proportion ± margin of error

The margin of error can be calculated using the formula:

Margin of Error = critical value * standard error

where the critical value is determined based on the desired confidence level and the standard error is calculated as:

Standard Error = \(\sqrt{((p * (1 - p)) / n)}\)

Given that the sample proportion (p) is 0.81 and the sample size (n) is 500, we can calculate the confidence intervals.

a. 90% Confidence Interval:

To find the critical value for a 90% confidence interval, we need to determine the z-score associated with the desired confidence level. The z-score can be found using a standard normal distribution table or calculator. For a 90% confidence level, the critical value is approximately 1.645.

Margin of Error = \(1.645 * \sqrt{(0.81 * (1 - 0.81)) / 500)}\)

≈ 0.0323

Confidence Interval = 0.81 ± 0.0323

≈ (0.7777, 0.8423)

Therefore, the 90% confidence interval for the population proportion is approximately (0.7777, 0.8423).

b. 95% Confidence Interval:

For a 95% confidence level, the critical value is approximately 1.96.

Margin of Error = \(1.96 * \sqrt{(0.81 * (1 - 0.81)) / 500)}\)

≈ 0.0363

Confidence Interval = 0.81 ± 0.0363

≈ (0.7737, 0.8463)

Thus, the 95% confidence interval for the population proportion is approximately (0.7737, 0.8463).

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Answer:

Since the amount dropped to 1/2 of the initial amount over a period of 7 years, you can assume the half-life is 7 years.

m(t) = m0 (0.5)t/7,

t = years elapsed from the time the amount was m0

In grams,

m(t) = 8 (0.5)t/7

m(8) = 8 (0.5)8/7 g ≅ ? g

Step-by-step explanation:

The procedure which is deemed to be the type of sample in discuss is; representative sample are free of bias.

Representative sampling and random sampling are two techniques used to help ensure data is free of bias. A representative sample on the other hand is a group or set chosen from a larger statistical population according to specified characteristics. A random sample is a group or set chosen in a random manner from a larger population.

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C. The image shows two right triangles with one pair of congruent leg and congruent hypotenuse. Therefore, the congruence is HL.

D. This shows two right triangles that share one side (a leg) and have the other leg also congruent to each other. Therefore, their congruence is LL and SAS (leg, right angle, leg).

E. This shows two triangles with a pair of congruent sides, a pair of congruent angles, and they have a common side, therefore congruent since they are the same.

So, their congruence is SAS.

F. This shows two right triangles sharing the same hypotenuse, and with a pair of congruent angles. Therefore, their congruence is HA.

G. This shows two right triangles with congruent hypotenuses, one congruent pair of legs, and two congruent pairs of angles (if we include the right angle).

So, their congruence is HA and HL.

H. The two triangles here have a right angle, a pair of congruent sides, and a pair of angles that are opposite by vertex, so they are also congruent angles.

Therefore, their congruence is ASA.

Answer:

1/125

Step-by-step explanation:

=1/5*5*5

=1/5^3

=1/125

Using exponent rules, when a number is raised to a negative exponent, change the exponent to positive and make a fraction with 1 as the numerator and the number as the denominator:

5^-3 = 1/(5^3)

Answer:

23

Step-by-step explanation:

The statements that correctly describe the data are:

a. The data shows an outlier at 70 degrees.

d. The data shows that, as the temperature at the track increases, more people visit.

e. The data shows that the temperature at the walking track depends on the number of people who visit.

In summary, the data includes an outlier at 70 degrees, indicating a data point that significantly deviates from the other values.

Additionally, the data suggests a positive relationship between temperature and the number of people visiting the track. As the temperature increases, more people tend to visit.

Furthermore, the data implies a relationship between the temperature at the walking track and the number of people who visit. The temperature is influenced by the number of visitors.

It's important to note that the given statements do not mention anything about the nature of the association being linear or nonlinear.

Therefore, statement c, which claims a positive nonlinear association, cannot be determined based on the information provided.

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Answer:

D

\(r - p \leqslant 150\)

Step-by-step explanation:

Break it up into two trapezoids as shown

area = trap1 + trap2

      = 2 *   (7+3) / 2   + 3 *  ( 7 + 3) / 2 = 10 + 15 = 25 units^2

Answer:

E) C and D. is the answer to ur q

Answer:

B

Step-by-step explanation:

Answer:

6 yards 4 feet 1 inch

Step-by-step explanation:

First, we have the equation:

4 yards 2 feet 7 inches + 2 yards 1 foot 6 inches

Now, let's add each:

6 yards

3 feet

13 inches

But 13 inches would be 1 foot and 1 inch, so now we have plus 1 foot and replace the inches with just 1 inch:

6 yards

4 feet

1 inch

There you have it!

Happy learning!

--Applepi101

Answer:

7 yds  1 ft 1 inch

Step-by-step explanation:

 4 yards 2 feet 7 inches

+ 2 yards 1 foot 6 inches

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

 6 yards  3 feet  13 inches

13 inches is more than 12 inches (  =1  ft) so subtract 12 inches and add 1 ft

6 yards  3 feet  13 inches

             =1 ft      - 12 inches

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

6 yds   4 ft    1 inches

4 ft is more than  3f which is 1 yd  ( subtract 3 ft and add 1 yd)

6 yds   4 ft    1 inches

+1 yd  - 3ft

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

7 yds  1 ft 1 inch

Let's check each student's response for different values of x:Student 1: 7x - 5 > 3x - 2

For x = 1:

7(1) - 5 =

2,

3(1) - 2 = 1

=> 2 > 1 (True)

For x = 2: 7(2) - 5

= 9, 3(2) - 2

= 4

=> 9 > 4 (True)

Student 2: 6x - 3 > 5x + 4

For x = 1:

6(1) - 3

= 3,

5(1) + 4

= 9

=> 3 is not greater than 9 (False) Student 3:

5x + 1 > 3x + 2

For x = 1:

5(1) + 1 = 6,

3(1) + 2 = 5

=> 6 is greater than 5 (True)

For x = 2: 5(2) + 1 = 11,

3(2) + 2 = 8

=> 11 is greater than 8 (True)

Student 4: 4x + 8 > 5x + 2

For x = 1: 4(1) + 8

= 12,

5(1) + 2 = 7

=> 12 is greater than 7 (True)

For x = 2: 4(2) + 8

= 16,

5(2) + 2 = 12

=> 16 is greater than 12 (True)

Therefore, Student 1 has the correct response to Mrs. Mueller's inequality because their inequality is true for all values of x.

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Answer:

.27

Step-by-step explanation:

Answer:

12π

Step-by-step explanation:

→ Write down the formula

π × r² × h

→ Identify variables

r = 2, h = 3

→ Substitute in the numbers

π × 2² × 3

→ Simplify

12π

1. The Taylor polynomial at x = -0.34. 10^-5 ≤ |Pn(-0.34)| ≤ 0.34^n / n!

2. The minimum order of the Taylor polynomial is n = 4.

1. To find the minimum order of the Taylor polynomial centered at 0 for cos(x) required to approximate cos(-0.34) with an absolute error no greater than 10^-5, we can use the error bound formula for Taylor polynomials.

The error bound formula states that for a function f(x) and its nth-degree Taylor polynomial Pn(x) centered at a, the absolute error between f(x) and Pn(x) is bounded by:

|f(x) - Pn(x)| ≤ M * |x - a|^n / (n!)

where M is the maximum value of the (n+1)th derivative of f(x) on the interval of interest.

For cos(x), the maximum value of the (n+1)th derivative is 1 for all n.

Let's solve the inequality:

10^-5 ≤ |cos(-0.34) - Pn(-0.34)| ≤ |0 - Pn(-0.34)|

Since cos(0) = 1, we only need to consider the error of the Taylor polynomial at x = -0.34.

10^-5 ≤ |Pn(-0.34)| ≤ 0.34^n / n!

2. To find the minimum value of n, we can start with a small value of n and increase it until the right side of the inequality is less than or equal to 10^-5.

Let's calculate the right side for n = 1, 2, 3, ...

For n = 1: 0.34^1 / 1! ≈ 0.34

For n = 2: 0.34^2 / 2! ≈ 0.057

For n = 3: 0.34^3 / 3! ≈ 0.006

For n = 4: 0.34^4 / 4! ≈ 0.001

We can see that for n = 4, the right side is less than 10^-5.

Therefore, the minimum order of the Taylor polynomial is n = 4.

So, the answer is n = 4.

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Answer and Step-by-step explanation:

First equation:

y = -4x + 9

This equation has a negative slope (a slope that does downwards) and has a positive y-intercept (at (0, 9)).

Second Equation:

y = -4x - 4

This equation is parallel to the first line because it has the same slope. It also has a negative y-intercept (at (0, -4)).

#teamtrees #PAW (Plant And Water)

Answer:

Step-by-step explanation:

£2.25 X 60 = £135

135 - 125= £10

£10/125 x 100 = 8%

Answer: A. 1120 in³

Step-by-step explanation:

the formula for the volume of the pyramid: \(V=\frac{1}{3} lwh\)

l (length)=16

w (width)=14

h (height)=15

     \(V=\frac{1}{3} lwh\)

     \(V=\frac{1}{3} (16)(14)(15)\)

     \(V=\frac{1}{3} 3360\)

     \(V=1120\)

Answer:

The answer is option A.

Step-by-step explanation:

Hey there!!

Given,

Length (l)= 16 in.

breadth (b)= 14 in.

height (h)= 15 in.

we have,

\(v. of \: pramid = \frac{1}{3} \times a \: of \: base \times height\)

or, volume = {1/3 × (16×14)×15} cubic inch.

by simplifying it we get,

The volume is 1120 cubic inch.

Hope it helps..

Answer:

(x−5)^2+(y+6)^2=9

Step-by-step explanation:

The best description about the sampling distribution of means for the student's sample is mu x = 45, sigma x = 3.75, shape of distribution unknown.

Sampling distribution is defined as the probability distribution of a statistical measure which is got by the repeated sampling of a specific population.

Given that,

μ = 45 minutes and σ = 15 minutes

Sampling distribution of means for the student's sample is :

μₓ = μ = 45

σₓ = σ / √n, where n is the sample size.

σₓ = 15 / √(16) = 15 / 4 = 3.75

Now since the sample size 16 which is less than 30, shape of the distribution is unknown.

Hence the sampling distribution is, μₓ = 45, σₓ = 3.75, and the shape of distribution is unknown.

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a.

= (5x)² - 3 = 25x² - 3

- 5f(x) = 5(x² - 3) = 5x² - 15

b.

- f(x - 2) = (x - 2)² - 3 = x² - 4x + 1

- f(x) - f(2) = (x² - 3) - (2² - 3) = x² - 3 - 1 = x² - 4

a. To calculate f(5x), we substitute 5x into the function f(x) and simplify the expression.

  f(5x) = (5x)² - 3 = 25x² - 3

  To calculate 5f(x), we multiply the function f(x) by 5.

  5f(x) = 5(x² - 3) = 5x² - 15

b. To calculate f(x - 2), we substitute (x - 2) into the function f(x) and simplify the expression.

  f(x - 2) = (x - 2)² - 3 = x² - 4x + 4 - 3 = x² - 4x + 1

  To calculate f(x) - f(2), we evaluate f(x) and f(2) separately and then find their difference.

  f(x) = x² - 3

  f(2) = 2² - 3 = 4 - 3 = 1

  f(x) - f(2) = (x² - 3) - (2² - 3) = x² - 3 - 1 = x² - 4

For the difference quotient of f(x) = -7x² - 5x + 9, we can calculate it as follows:

Difference quotient = [f(x + h) - f(x)] / h

Expanding the function and substituting into the difference quotient formula, we have:

[f(x + h) - f(x)] / h = [-7(x + h)² - 5(x + h) + 9 - (-7x² - 5x + 9)] / h

Simplifying and expanding further:

= [-7(x² + 2hx + h²) - 5x - 5h + 9 + 7x² + 5x - 9] / h

= [-7x² - 14hx - 7h² - 5x - 5h + 9 + 7x² + 5x - 9] / h

= [-14hx - 7h² - 5h] / h

= -14x - 7h - 5

The difference quotient of f(x) = -7x² - 5x + 9 is -14x - 7h - 5.

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Answer:

\(9n - 12 \rightarrow \frac{1}{3} (9n - 12) \\ = \frac{3}{3} (3n - 4) \\ l = 3n - 4\)

The perimeter of an equilateral triangle is 9n - 12. Write an expression to represent the length of each side.

\( = \frac{9n-12}{3} \\ = \frac{3(3n - 4)}{3} \\ = 3n - 4\)

The length of each side of the triangle is 3n-4.

Hope it helps!!

Answer:

No

Step-by-step explanation: they're is no constants they are both coeffitient because they are both numbers infront of the variable

The probability that the sample mean would be less than 127 wpm if 84-speed typists are randomly selected is: 0.033625

What is Mean?

In statistics, in addition to the mode and median, the mean is one of the measures of central tendency. Simply said, the mean is the average of the values in the given collection. It indicates that values in a certain data collection are distributed equally. The three most often employed measures of central tendency are the mean, median, and mode. The total values provided in a datasheet must be added, and the sum must be divided by the total number of values in order to get the mean. When all of the values are organized in ascending order, the Median is the median value of the provided data. While the number in the list that is repeated a maximum of times is the mode.

We have

\(\mu=124\)  

\(\sigma=15\)

n=84

We want to find the probability that the sample mean would be greater than 127 WPM is

\(\begin{aligned}P(\bar{X} > 127) &=P\left(\frac{\bar{X}-\mu}{\sigma / \sqrt{n}} > \frac{127-124}{15 / \sqrt{84}}\right) \\&=P(Z > 1.83)\end{aligned}\)

\(=0.033625\)

The required probability is 0.0336.

Hence, The probability that the sample mean would be less than 127 wpm if 84-speed typists are randomly selected is: 0.033625

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