A recent article in a national newspaper claimed that 40% of all students in US colleges drink some alcoholic beverages at least once a week. A students association argued that actually fewer students arink alcohol. In support of its claim the student association has sampled at random 500 students and found that only 180 of them stated that they do drink at least once a week. What is the (claimed) population proportion of students who drink at least once a week? p= By the CLT the distribution of the sample proportion p ​ is the normal distribution with Mean: μ j ​ =p and standard deviation σ j ​ = n pq ​ ​ . Caiculate the values of these parameter assuming that the claimed population proportion in (a) is correct, μ p ​ = and σ 5 ​ = c) Calculate the (observed) sample proportion of those who said "Yes" (that they do drink at least once a week). p ^ ​ 0 ​ = 500 180 ​ =0.36 p ˙ ​ at ​ = n x ​ =0.36 d) Use the CLT and (b) to find the probability to observe such, or smaller, sample proportion if the claim in (a) were to be correct. Pr( p ​ ≤ P ^ ade ​ )= e) is the result in (d) supports or refutes the claim in (a)? Why?

The correct step in the solution of the equation [tex]\sqrt[4]{}[/tex](2m+1-2) = 1 is option C: [tex]\sqrt[4]{}[/tex](2m-1) = 1.

An equation is a mathematical statement that indicates that two expressions are equal. It consists of two sides separated by an equal sign (=). The expressions on either side of the equal sign can contain variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. The goal of solving an equation is to find the value of the variable that makes the equation true. Equations are used in many areas of mathematics, as well as in physics, engineering, and other sciences, to model and solve problems.

We can start solving the equation [tex]\sqrt[4]{}[/tex](2m+1-2) = 1 by simplifying the left side of the equation first. We have:

[tex]\sqrt[4]{}[/tex](2m+1-2) = 1

[tex]\sqrt[4]{}[/tex](2m-1) = 1

²(2√(2m-1)) = 1 (using the fact that 4 = 2²)

2sqrt(2m-1) = 0 (taking the square root of both sides)

At this point, we can see that the equation simplifies to 2*√(2m-1) = 0, which means that √(2m-1) = 0 (since 2 ≠ 0). Therefore, we can solve for m by squaring both sides:

√(2m-1) = 0

(√(2m-1))² = 0²

2m-1 = 0

2m = 1

m = 1/2

Therefore, the correct step in the solution of the equation [tex]\sqrt[4]{}[/tex](2m+1-2) = 1 is option C: [tex]\sqrt[4]{}[/tex](2m-1) = 1.

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Answer:

Step-by-step explanation:

The inequality is x > 45.

This is so because the dot is not shaded which tells us that the value of x is not equal to 45.

Also, the direction of the arrow is going to the right on the positive side of the number line which means it's greater than 45.

Lisa's situation could be represented by just reversing the arrow on the above graph.

Hope this helps! :)

Water depth at 10:10 am is approximately 4.34 feet.

Given data:The low tide occurred at 5:40 am with a water depth of 1.25 feet. Six hours and 2 minutes later, the high tide occurred with a water depth of 6.82 feet.Tide is the periodic rise and fall of sea level due to the gravitational pull of the Moon and the Sun on the Earth. This motion can be modeled by sinusoidal functions.The general form of the sine function is given as f(x) = a sin bx + c, where a is the amplitude, b is the period, and c is the vertical shift.To find all components needed to write a model for this scenario since the first low tide:First, we need to calculate the amplitude and period of the sine wave, as follows:Given that the low tide occurred at 5:40 am with a water depth of 1.25 feet.Similarly, the high tide occurred with a water depth of 6.82 feet at 11:42 am, which is 6 hours and 2 minutes after the low tide. Therefore, the period of the wave is 12 hours and 4 minutes or 12.067 hours.Amplitude is given as half the difference between the maximum and minimum values of the wave, which is 2.285 feet.Writing the model for the scenario since the first low tide is given as:f(x) = 2.285 sin ((2π/12.067) x) + cSince the water depth at low tide is 1.25 feet, the vertical shift is given as 3.065 feet.f(x) = 2.285 sin ((2π/12.067) x) + 3.065Now, we need to find the water depth at 10:10 am.To do that, we substitute x = 4.5 (since 5:40 am to 10:10 am is a period of 4.5 hours) in the above equation, and we get:f(4.5) = 2.285 sin ((2π/12.067) × 4.5) + 3.065≈ 4.34 feetTherefore, the water depth at 10:10 am is approximately 4.34 feet.

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Answer:

A. area of larger rectangle: 18x^2

area of smaller rectangle: 8x^2

B. area of shaded region: 10x^2

Step-by-step explanation:

Larger rectangle: A = lw = (6x)(3x) = 18x^2

Smaller rectangle: A = lw = (4x)(2x) = 8x^2

The shaded region = larger rectangle - smaller rectangle

=> 18x^2 - 8x^2 = 10x^2

The area of the composite figure is calculate to be 96.07 square in

The area of the composite figure is solved by dividing the figure into simpler figures

We have to divide the given figure into 3 figures

2 rectangles and a quarter of a circle

Area of the first rectangle

= base * height

= 7 * 7

= 49 square in

Area of the second rectangle

= base * height

= 5 * 8

= 40 square in

Area of the quarter of a circle

= 1/4 * π r²

= 0.25 * π * 3²

= 7.07 square in

Area of the composite figure

= 49 square in + 40 square in + 7.07 square in

= 96.07 square in

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The sunflower's height is increasing at an average rate of 13.045 inches per week over the third week.

The average rate of change of a function over an interval is the slope of the secant line that passes through the two endpoints of the interval. Mathematically, if we have a function f(x) and an interval [a,b], the average rate of change of f(x) over [a,b] is given by:

average rate of change = (f(b) - f(a))/(b - a)

For our problem, the function is f(t) = 5(1.7)ˣ, and we need to find the average rate of change over different time intervals. Let's consider each interval separately:

The average rate of change over the [0,1] interval is:

average rate of change = (f(1) - f(0))/(1 - 0) = (5(1.7)¹ - 5(1.7)⁰)/(1 - 0) = 4.5

Therefore, the sunflower's height is increasing at an average rate of 4.5 inches per week over the first week.

The average rate of change over the [1,2] interval is:

average rate of change = (f(2) - f(1))/(2 - 1) = (5(1.7)² - 5(1.7)¹)/(2 - 1) = 7.65

Therefore, the sunflower's height is increasing at an average rate of 7.65 inches per week over the second week.

The average rate of change over the [2,3] interval is:

average rate of change = (f(3) - f(2))/(3 - 2) = (5(1.7)³ - 5(1.7)²)/(3 - 2) = 13.045

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Mrs. Irfan needs to buy 10,000 tiles to cover the floor of her balcony.

To determine the number of tiles Mrs. Irfan needs to cover her balcony, we need to first calculate the area of each tile. The area of each tile can be determined by multiplying its length and breadth:

Area of each tile = Length × Breadth = 20 cm × 10 cm = 200 cm²

Next, we need to calculate the total area of the balcony, which is given as 200 cm². However, we need to convert this to square centimeters since we have determined the area of each tile in square centimeters. We can do this by multiplying 200 cm² by 10,000 (1 square meter is equal to 10,000 square centimeters):

Total area of balcony in square centimeters = 200 cm² × 10,000 = 2,000,000 cm²

To determine the number of tiles Mrs. Irfan needs, we can divide the total area of the balcony by the area of each tile:

Number of tiles needed = Total area of balcony ÷ Area of each tile

= 2,000,000 cm² ÷ 200 cm²

= 10,000 tiles

Complete question:

Mrs. Irfan wants to use tiles of a length of 20 cm and a breadth of 10 cm to cover the floor of her balcony. The area of the balcony is 200 cm^2. how many tiles does she need to buy(1sq m = 10,000 Sq cm)?​

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The mean of the given frequency distribution is 587.12.

What is frequency distribution?

In frequency tables or charts, frequency distributions are displayed. The actual number of observations that fall into each range can be seen in frequency distributions, as well as the proportion of observations that do.

We are given a frequency distribution table.

We know that the mean is the average of sum of all the values.

So, we first get the values as :

⇒ 610 * 12 = 7320

⇒ 537 * 6 = 3222

⇒ 597 * 10 = 5970

⇒ 572 * 14 = 8008

⇒ 590 * 9 = 5310

⇒ 606 * 6 = 3636

Now, on adding all the values, we get

⇒ 7320 + 3222 + 5970 + 8008 + 5310 + 3636

⇒ 33466

So,

⇒ Mean =  33466 ÷ 57

⇒ Mean = 587.12

Hence, the mean of the given frequency distribution is 587.12.

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The new coordinates after the scale factor comes into the scenario is:

1. (-12,3), (-6,9), (0, -6), (-15, -6)

2. (1, -4), (6,4), (7, -2)

3. (4,3), (12,12), (12,8), (4,8)

4. (3, -6), (9, -3), (15, -6), (9, -9)

5. (-14, -8), (-8, -6), (-6, -12), (-12, -14)

6. (-1,3), (2,2), (2,1), (-1,0)

During the process of dilatation, an object must be reduced in size or changed. It is a transformation that uses the given scale factor to shrink or expand the objects. The image is the new figure that forms as a result of dilatation, whereas the pre-image is the original figure. There are two kinds of dilation:

A rise in an object's size is referred to as expansion.

Contraction is the term for a reduction in size.

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Answer:

5,040

Step-by-step explanation:

There are a total of 5,040 interesting numbers, which can be calculated using basic combinatorics. To calculate this, the first step is to count the total number of 10-digit numbers with distinct digits. Since there are 10 distinct digits (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9), there are 10! (10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) possible numbers with distinct digits. The next step is to count the number of 10-digit numbers divisible by 11,111. To do this, we need to find the number of 10-digit numbers divisible by 11. Since 11 is a prime number, there are 11 possible combinations of 10-digit numbers divisible by 11. For example, 0XXXXXXXXX, 1XXXXXXXXX, 2XXXXXXXXX, etc. So, there are 11 x 10! (110,000) possible 10-digit numbers divisible by 11. Finally, to find the number of interesting numbers, we need to subtract the number of 10-digit numbers divisible by 11,111 from the total number of 10-digit numbers with distinct digits. This gives us 10! - 11 x 10! = 5,040. Therefore, there are a total of 5,040 interesting numbers.

Answer:

To find the number of interesting numbers, we need to find the number of 10-digit numbers that are divisible by 11,111 and have distinct digits.

A number is divisible by 11,111 if and only if the alternating sum of its digits is divisible by 11. For example, 27,183,642 is divisible by 11,111 because 2 - 7 + 1 - 8 + 3 - 6 + 4 - 2 = -3, which is divisible by 11.

Since we want the number to have distinct digits, we can choose any 5 digits from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and arrange them in any order. There are 10 choices for the first digit, 9 choices for the second digit (since we can't repeat the first digit), 8 choices for the third digit, and so on, giving us a total of 10 x 9 x 8 x 7 x 6 = 30,240 possible 5-digit numbers with distinct digits.

To form a 10-digit number, we can repeat the 5 digits in any order. There are 5! = 120 ways to arrange the 5 digits, so there are 120 different 10-digit numbers that can be formed from each 5-digit number.

Therefore, the total number of interesting numbers is:

30,240 x 120 = 3,628,800

So there are 3,628,800 interesting numbers that are 10-digit numbers divisible by 11,111 with distinct digits.

Answer: From the graph, we see that Anna needs to sell 300 pre-sale tickets and 100 at-the-door tickets to reach her goal. Therefore, she needs to sell 100 at-the-door tickets to make her goal. Answer: 100.

Step-by-step explanation:

Answer:

£756

Step-by-step explanation:

To fill the tank completely, Mr. Patel needed:

500 liters initially + (total capacity of the tank - 500 liters) = 500 + (tank capacity - 500) = tank capacity

To find the tank capacity, we need to know how many liters of oil Mr. Patel bought initially:

Total cost of oil bought initially = 900

Cost per liter of oil = £0.60

Number of liters of oil bought initially = 900 / 0.60 = 1500

Therefore, the tank capacity is:

500 + (tank capacity - 500) = 1500

Tank capacity - 500 = 1000

Tank capacity = 1500 liters

At the beginning of February, Mr. Patel had 300 liters of oil left in his tank, so he needed to buy:

1500 - 300 = 1200 liters of oil

The cost of oil had increased by 5%, so the cost per liter was:

£0.60 + (£0.60 x 0.05) = £0.63

Therefore, the total cost of the oil in February was:

1200 x £0.63 = £756

Answer:

2

Step-by-step explanation:

2

Values of 12 and 13 are 76° and 63° respectively.

A polygon with three sides and three angles is triangle.It is the simplest polygon and can be classified based on its sides and angles. Triangles are used in various fields, including mathematics, engineering, architecture, and art. They are also used to represent stability, strength, and balance in symbols and logos.

Given:-∠F = 104°

Let we assume ∠12 = x

therefore,

∠12 + ∠F = 180°

x + 104 = 180°

x = 180 - 104

x = 76°

therefore , ∠12 is 76°.

now ,

∠D = 41°

Sum of all angles of triangle is 180°

so,

∠D + ∠12 + ∠13= 180°

41 + 76 + ∠E = 180

117 + ∠E = 180

∠E = 180 - 117

∠E = 63°

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The volume of a liquid can also be compared to the volume of a solid, as liquids and solids both occupy space.

What is Volume ?

Volume is a measure of the amount of space occupied by an object or substance in three-dimensional space. It is the amount of space that a solid, liquid, or gas occupies.

Liquid volume is the amount of space occupied by a liquid.

The units of liquid volume are typically liters, milliliters, gallons, or fluid ounces.

Liquid volume can be measured using a graduated cylinder or other measuring tools.

The volume of a liquid can be affected by changes in temperature and pressure.

The volume of a liquid can be calculated by multiplying its height, width, and length.

The density of a liquid can also affect its volume, as denser liquids will occupy less space than less dense liquids.

The volume of a liquid can be converted to other units of measurement using conversion factors.

Therefore, The volume of a liquid can also be compared to the volume of a solid, as liquids and solids both occupy space.

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The area of rectangle DAEF is 40 square units.

Since rectangle ABCD is similar to rectangle DAEF, their corresponding sides are proportional.

Let the length of rectangle DAEF be x.

Then, we have the following ratios:

AB/DA = EF/DA (corresponding sides of similar rectangles are proportional)

10/4 = x/4 (substituting AB=10 and AD=4)

Solving for x, we get:

x = 40/10 = 4

Therefore, the length of rectangle DAEF is 4.

Now, the area of rectangle DAEF is:

Area = length x width

Area = 4 x 10 = 40 square units.

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The calculated value of x in the right triangle is 34.9 degrees

From the question, we have the following parameters that can be used in our computation:

The right triangle

The value of x can be calculated using

tan(x) = 4.0/5.8

Evaluate the quotient

So, we have

tan(x) = 0.6897

Take the arctan of both sides

So, we have

x = 34.9 degrees

Hence, the value of x is 34.9 degrees

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Answer:

Step-by-step explanation:

In a parallelogram, the two opposite sides are parallel and equal

WR = 2 (6x - 7.7)

221 = 2( 6x - 7.7)

Answer:

Step-by-step explanation:

93+23 then u takeaway the whole number to 93

Answer:

The answer is 70.

Step-by-step explanation:

If you subtract 23 from 93, you will get 70.

Using structural induction, the length of all elements x of B is an integer power of 2, for all n∈N, sumeven(n) = n(n+1), for all elements u of D, u starts and ends with the same character.

a) Basis Step: For the string 0, the length is 1 which is equal to 2^0, and for the string 1, the length is also 1 which is equal to 2^0. Hence, the property holds for the basis step.

Recursive Step: Assume that the length of the binary string x is an integer power of 2. Then, the length of xx is twice the length of x which is also an integer power of 2. Therefore, the length of xx is an integer power of 2. Hence, the property holds for the recursive step.

Therefore, by structural induction, the length of all elements x of B is an integer power of 2.

(b) Basis Step: For n = 0, sumeven(n) = 0 and n(n+1) = 0(0+1) = 0. Hence, the property holds for the basis step.

Recursive Step: Assume that for some n∈N, sumeven(n) = n(n+1). We need to show that sumeven(n+1) = (n+1)(n+2).

Using the recursive step of the definition of sumeven, we have:

sumeven(n+1) = sumeven(n) + (2(n+1)+2)

= n(n+1) + 2n + 4

= n^2 + 3n + 2

= (n+1)(n+2)

Hence, the property holds for the recursive step.

Therefore, by structural induction, for all n∈N, sumeven(n) = n(n+1).

(c) Basis Step: For the strings 0 and 1, the property holds since they start and end with the same character.

Recursive Step: Assume that for some strings u, v, and w, u and w start and end with the same character. We need to show that the string wxw also starts and ends with the same character.

Since u and w start and end with the same character, we can write u = axa and w = byb for some characters a, b, x, and y. Then, the string wxw can be written as axaybybaxa which starts and ends with the same character a.

Similarly, we can write u = axa and w = byb for some characters a, b, x, and y such that a ≠ b. Then, the string wxw can be written as axaybybaxa which starts and ends with the same character a.

Therefore, by structural induction, for all elements u of D, u starts and ends with the same character.

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The standard value that corresponds to a sample mean age of 60 years is 0.

We can use the formula for the z-score (standardized value) to find the answer:

z = (x - μ) / (σ / sqrt(n))

where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.

In this case, we have:

x = sample mean age of the 400 trees

μ = population mean age of trees in the large park = 60 years

σ = population standard deviation of tree ages in the large park = 2.2 years

n = sample size = 400

We don't know the value of x, but we do know that the distribution of sample means is approximately normal, with a mean of μ = 60 and a standard deviation of σ / sqrt(n) = 2.2 / sqrt(400) = 0.11.

So, we want to find the standardized value for a sample mean that is 0 standard deviations away from the population mean, which means:

z = (x - 60) / 0.11 = 0

Solving for x, we get:

x - 60 = 0

x = 60

Therefore, the standardized value that corresponds to a sample mean age of 60 years is:

z = (x - μ) / (σ / sqrt(n))

z = (60 - 60) / 0.11

z = 0

So the answer is 0.

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Answer:

c ≈ 6.4 in

Step-by-step explanation:

using Pythagoras' identity in the right triangle.

the square on the hypotenuse (c) is equal to the sum of the squares on the other 2 sides, that is

c² = 4² + 5² = 16 + 25 = 41 ( take square root of both sides )

c = [tex]\sqrt{41}[/tex] ≈ 6.4 in ( to the nearest tenth )

The value of the probability that Y is less than or equal to 9 is approximately 0.893

Given that the moment generating function:

M(t) = (pe⁺ + q)ⁿ

And also

q = 1 - p

When M(t) = (0.8e⁺ + 0.2)¹⁰ and M(t) = (pe⁺ + q)ⁿ are compared, we have

n = 10

p = 0.8

q = 0.2

To find P(Y ≤ 9), we can use the cumulative distribution function (CDF) for the binomial distribution:

[tex]F(k) = P(Y \le k) = \sum\limits^k_{i=0}\left[\begin{array}{c}n&i\end{array}\right] p^i q^{n-i}[/tex]

In this case, we want to find P(Y ≤ 9), so we can evaluate the CDF at k=9:

So, we have

[tex]P(Y \le 9) = \sum\limits^9_{i=0}\left[\begin{array}{c}10&i\end{array}\right] 0.8^i * 0.2^{n-i}[/tex]

Using a calculator to evaluate this sigma notation, we find that

P(Y ≤ 9) ≈ 0.89263

Approximate

P(Y ≤ 9) ≈ 0.893

Therefore, the probability that Y is less than or equal to 9 is approximately 0.893

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Complete question

If Y has a binomial distribution with n trials and probability of success p, the moment-generating function for Y is M(t) = (pe⁺ + q)ⁿ, where q = 1 − p.

If Y has moment-generating function M(t) = (0.8e⁺ + 0.2)¹⁰, what is P(Y ≤ 9)?

Answer:

please mark as brainliest

Answer:

Using Newton's Method, the points of intersection between f(x) = x^6 and g(x) = cos(x) are approximately (0.8241, f(0.8241)) and (2.3111, f(2.3111)), where f(x) = x^6.

To find the points of intersection of the graphs of f(x) = x^6 and g(x) = cos(x), we can solve the equation h(x) = f(x) - g(x) = x^6 - cos(x) = 0.

Explanation:

We will use Newton's Method to approximate the x-value(s) of intersection. The formula for Newton's Method is:

x_n+1 = x_n - f(x_n)/f'(x_n)

where x_n is the nth approximation of the root, f(x_n) is the function evaluated at x_n, and f'(x_n) is the derivative of the function evaluated at x_n.

Let h(x) = x^6 - cos(x), then

h'(x) = 6x^5 + sin(x)

Now we need to choose a starting value for x. By graphing the two functions, we can see that there are two points of intersection in the interval [0,1]. Let's choose x = 0.5 as our starting value.

x_0 = 0.5

x_1 = x_0 - h(x_0)/h'(x_0) = 0.5352

x_2 = x_1 - h(x_1)/h'(x_1) = 0.8656

x_3 = x_2 - h(x_2)/h'(x_2) = 0.8249

x_4 = x_3 - h(x_3)/h'(x_3) = 0.8241

Thus, the approximate value of the first intersection point is x = 0.8241.

Now we need to find the second intersection point. By graphing the two functions, we can see that there is another intersection point in the interval [2,3]. Let's choose x = 2.5 as our starting value.

x_0 = 2.5

x_1 = x_0 - h(x_0)/h'(x_0) = 2.3214

x_2 = x_1 - h(x_1)/h'(x_1) = 2.3111

x_3 = x_2 - h(x_2)/h'(x_2) = 2.3111

Thus, the approximate value of the second intersection point is x = 2.3111.

Therefore, the points of intersection of the two graphs are approximately (0.8241, f(0.8241)) and (2.3111, f(2.3111)), where f(x) = x^6.

Hope this helps you in some way! I'm sorry if it doesn't. If you need more help, ask me! :]

As a result, the simplified expressiοn is (1/4)m(-3)p. by subtracting the apprοpriate expοnents frοm 3 and then dividing it by 12.

A variable in mathematics is a symbοl οr letter that designates a number that is subject tο variatiοn οr change. Mathematical expressiοns and fοrmulae that can be sοlved tο determine the value οf a variable are written using variables. A, B, C, and οther symbοls are frequently used tο denοte variables, including x, y, and z.

Numerοus different types οf quantities, including integers, functiοns, vectοrs, matrices, and οthers, can be represented by them. X and Y are factοrs in the equatiοn y = 2x + 1, fοr instance. We can determine the cοrrespοnding number οf y by substituting a value fοr x.

given

By dividing 3 by 12 and taking away the cοrrespοnding expοnents οf r and p, we can first simplify the numeratοr οf the expressiοn.

[tex](3m^{(-3)}r^{(4)}p^{(2)})/(12r^{(4)}) = (1/4)m^{(-3)}r^{(4-4)}p^{(2)}[/tex]

Even mοre simply put, we have:

[tex](1/4)m^{(-3){p^{(2)}[/tex]

As a result, the simplified expressiοn is (1/4)m(-3)p. by subtracting the apprοpriate expοnents frοm 3 and then dividing it by 12.

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Answer: Based on the description of the figure, the first three parts of the rectangle are shaded dark to represent the fraction 3/8, and the next three parts are shaded light to complete the fraction 3/4. The last two parts are not shaded.

To find the quotient of 3/4 divided by 3/8, we can use the rule that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of 3/8 is 8/3, so we have:

3/4 ÷ 3/8 = 3/4 × 8/3

To simplify this expression, we can cancel out a factor of 4 from the numerator and denominator of 3/4, and a factor of 3 from the numerator and denominator of 8/3. This gives us:

3/4 × 8/3 = (3 × 2)/(1 × 1) = 6

Therefore, the quotient of 3/4 divided by 3/8 is 6.

Step-by-step explanation:

Therefore , the solution of the given problem of triangle comes out to be   m∠B = 29.5 degrees , m∠C = 132.25 degrees and m∠D = 18.25 degrees.

Because a triangle has two or so more extra parts, it is a polygon. It has a straightforward rectangular shape. Only two of a triangle's three sides—A and B—can differentiate it from a regular triangle. Euclidean geometry produces a single area rather than a cube when boundaries are still not perfectly collinear. Triangles are defined by their three sides and three angles. Angles are formed when a quadrilateral's three sides meet. There are 180 degrees of sides on a triangle.

Here,

Angles B and D are congruent because triangle BCD is isosceles with basis BD. As a result, we can equalise their measurements and find x:

=> m∠B = m∠D

=> (5x + 4) = (x + 15)

=> 4x = 11

=> x = 11/4

Knowing x allows us to determine the size of each angle.

=> m∠B = 5x + 4 = 5(11/4) + 4 = 29.5 degrees

=> m∠D = x + 15 = (11/4) + 15 = 18.25 degrees

Angles B and D being congruent, we can determine what mC is as follows:

=> m∠C = 180 - m∠B - m∠D = 180 - 29.5 - 18.25 = 132.25 degrees

As a result, the triangle's angles are each measured in degrees as follows:

=> m∠B = 29.5 degrees

=>  m∠C = 132.25 degrees

=>  m∠D = 18.25 degrees

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The total number of cells after 9 hours is approximately 10671.

An exponential expression is a mathematical expression in which a number or variable is raised to a power, which is usually written as a superscript. The base is the number or variable being raised to a power, and the exponent is the number indicating how many times the base is multiplied by itself.

In the given question,

The initial growth of the bacterial cells can be modeled by an exponential function of the form:

N(t) = N0 * 3^(kt)

Where N(t) is the number of cells at time t, N0 is the initial number of cells, k is the growth rate, and t is the time elapsed.

Using the information given in the problem, we can find N0 and k as follows:

N0 = 1 (since the researchers started with one cell)

N(5) = 35

N(5) = N0 * 3^(5k) = 35

3^(5k) = 35

5k = log3(35)

k = log3(35)/5

Therefore, the function for the growth of bacterial cells is:

N(t) = 3^(t*log3(35)/5)

To find the total number of cells after 9 hours, we can substitute t=9 into the equation:

N(9) = 3^(9*log3(35)/5) ≈ 10671

Therefore, the total number of cells after 9 hours is approximately 10671.

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The measure of side AC is approximately 366.9 inches.

A triangle is a geometric shape that consists of three straight sides and three angles. It is one of the basic shapes in geometry and is often studied as a fundamental building block in mathematics, engineering, and physics. Triangles have several important properties, including:

The sum of the angles in a triangle is always 180 degrees.

The longest side of a triangle is opposite to the largest angle, and the shortest side is opposite to the smallest angle.

The area of a triangle can be calculated using the formula: area = 1/2 x base x height, where the base is one of the sides and the height is the perpendicular distance from the base to the opposite vertex.

Triangles can be classified based on the length of their sides and the measure of their angles. For example, a triangle with all three sides of equal length is called an equilateral triangle, while a triangle with two sides of equal length is called an isosceles triangle.

To find the measure of side AC, which is denoted by "b", we can use the Law of Cosines, which relates the lengths of the sides of a triangle to the cosine of one of its angles. Specifically, the Law of Cosines states that:

c² = a² + b² - 2ab cos(C)

where "a" and "b" are the lengths of the other two sides of the triangle, and "C" is the angle opposite to side "c".

In this case, we know that side AB has length "c" (290 inches), and side AC is denoted by "b". We also know that the angle opposite to side AB has measure 42 degrees. Therefore, we can write:

c² = a² + b² - 2ab cos(C)

290² = a² + b² - 2ab cos(42)

Simplifying this equation and solving for b, we get:

b² - 2ab cos(42) + (290² - a²) = 0

This is a quadratic equation in "b", which we can solve using the quadratic formula:

b = [2a cos(42) ± √((2a cos(42))² - 4(290² - a²)]/2

Simplifying this expression, we get:

b = a cos(42) ± √(a² cos²(42) - (290² - a²))

We don't know the value of "a", so we cannot find the exact value of "b". However, we can use the fact that "b" is a length of a triangle, so it must be positive. Therefore, we can discard the negative square root, and we get:

b = a cos(42) + sqrt(a² cos²(42) - (290² - a²))

Since "b" is positive, we can set the expression inside the square root to zero and solve for "a":

a² cos²(42) - (290² - a²) = 0

Simplifying and solving for "a", we get:

a = 290/sin(42)

Substituting this value of "a" back into the equation for "b", we get:

b = (290/sin(42)) cos(42) + sqrt((290/sin(42))² cos²(42) - 290² + (290/sin(42))²)

Simplifying this expression, we get:

b ≈ 366.9 inches

Therefore, the measure of side AC is approximately 366.9 inches.

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