Can someone help me solve this and explain it?

Step-by-step explanation:

(sinθ + cosθ)^2 + (sinθ − cosθ)^2

= (sin^2 θ + cos^2 θ + 2sinθ cosθ)+(sin^2 θ+cos^2 θ − 2sinθ cosθ)

= (1 + 2sinθ cosθ) + (1 − 2sinθ cosθ)

= 1 + 2sinθ cosθ + 1 − 2sinθ cosθ

= 1 + 1 = 2

Therefore, (sinθ+cosθ)^2 + (sinθ−cosθ)^2 = 2

a)30*20 b)30*30 d)25*32 e)25*30 are NOT good ways to use compatible numbers to estimate the number of cans in the crates.

And the best expressions that is the good ways to use compatible numbers to estimate the number of cans in the crates is 32*27

As given in the question,

There are 32 crates. Each crate holds 27 cans of soup.

i.e. Number of total crates = 32

Number of can of soups in each crate = 27

The total number of cans of soup in every crates = 32*27

The total number of cans of soup in every crates = 864

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a. 2x + 3y = 7

2y = 12x - 2

To solve this system of equations, we can begin by isolating one of the variables. Let's first isolate y in the second equation:

2y = 12x - 2

y = 6x - 1

Now we can substitute this expression for y into the first equation:

2x + 3(6x - 1) = 7

2x + 18x - 3 = 7

20x = 10

x = 1/2

Now that we know the value of x, we can substitute it back into the equation y = 6x - 1 to find the value of y:

y = 6(1/2) - 1

y = 3/2 - 1

y = -1/2

So the solution to this system of equations is (x, y) = (1/2, -1/2).

b. 2x - 9y = 12

3x + 18y = 18

To solve this system of equations, we can begin by using the first equation to solve for one of the variables. Let's solve for x:

2x - 9y = 12

2x = 9y + 12

x = (9y + 12)/2

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

3((9y + 12)/2) + 18y = 18

(27y + 36) + 18y = 18

45y = -18

y = -4/5

Now we can substitute this value of y back into the equation x = (9y + 12)/2 to find the value of x:

x = (9(-4/5) + 12)/2

x = (3 + 12)/2

x = 15/2

So the solution to this system of equations is (x, y) = (15/2, -4/5).

The standard form of the given number -2+√-8 is -2+2i

What is the imaginary number?

An imaginary number is a number that can be written in the form of a real number multiplied by the imaginary unit, denoted by the symbol "i". The imaginary unit, "i", is defined as the square root of -1, which is a number that when multiplied by itself results in -1.

The given number cannot be written in standard form, because √-8 is an imaginary number and cannot have a real value. The square root of a negative number is an imaginary number, and these numbers are denoted by the imaginary unit, i.

The imaginary unit is defined as the square root of -1.

The number -2+√-8 is equal to -2+i*2.

The standard form of a complex number is a+bi, where a and b are real numbers and i is the imaginary unit.

So, The standard form of the given number -2+√-8 is -2+2i

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The standard deviation in statistics is a measurement of how much a group of values can vary or be dispersed.

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There can be a maximum of 5 teams.Three persons will be excluded.

Mathematical expression is the combination of numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented using mathematical symbols. They can also be used to denote the logical syntax's operation order and other properties.

Considering that 23 pals gather to play a sport together.The participants must first be divided into teams. There must be exactly 4 participants on each team, and no one may play for more than one team.

The number of teams will be determined using the formulas below:

Total teams = 23 / 4 = 20 / 4 = 5.

Three persons will be left.

Therefore, there can be a maximum of 5 teams.Three persons will be excluded.

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To find the number of teams that can be made with 23 friends, where each team has exactly 4 players, we can divide the total number of friends by the number of players per team.

Number of friends = 23

Number of players per team = 4

To find the number of teams, we divide the number of friends by the number of players per team:

23 / 4 = 5 remainder 3

Since we cannot have a team with 3 players, we can only form complete teams of 4 players. Therefore, the maximum number of teams that can be formed is 5.

Therefore, the answer is 5 teams.

x = No. of girls

y = no. of boys

2y + y = 42

3y = 42

y = 14

no. of girls = 2 × 14 = 28

no of boys = 14

6 is the number of ways described when There are 3 children, 3 pets, and 3 favorite colors. We can represent the children, pets, and colors in a list and explained below .

Children:

1. Angela

2. Lisa

3. Susan

Pets:

1.Cat

2. Dog

3. Fish

Colors:

1.Blue

2.Green

3. Red

We can use the combination rule to find the number of ways to assign a child, a pet, and a color to each other. The combination rule states that if we have n items and want to choose k items at a time, the number of ways to do this is given by the formula:

n! / (k! (n-k)!)

Where n! denotes the factorial of n, meaning the product of all positive integers less than or equal to n.

In this case, we have 3 children, 3 pets, and 3 colors, and we want to assign one of each to each other. So we can use the formula to find the number of ways to do this:

3! / (1! (3-1)!) = 3! / (1! 2!) = 3 * 2 * 1 / (1 * 2) = 3

So there are 3 ways to assign a child, a pet, and a color to each other.

We can also use the permutation rule to find the number of ways to assign a child, a pet, and a color to each other. The permutation rule states that if we have n items and want to choose k items at a time and order matters, the number of ways to do this is given by the formula:

n! / (n-k)!

In this case, we have 3 children, 3 pets, and 3 colors, and we want to assign one of each to each other. So we can use the formula to find the number of ways to do this:

3! / (3-3)! = 3! / 0! = 3 * 2 * 1 / 1 = 6

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Since David drew a quadrilateral with opposite sides parallel, no right angles and all four sides are not the same length, the quadrilateral would be referred to as a scalene trapezoid.

In Geometry, there are different types of quadrilateral and these include the following:

Generally speaking, some of the properties of a scalene trapezoid include the following:

In conclusion, we can reasonably infer and logically deduce that David drew a scalene trapezoid.

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A student who is chosen at random has a 2/5 chance of playing neither instrument.

Probability is a number that expresses the likelihood or chance that a specific event will take place. Both proportions ranging from 0 to 1 and percentages ranging from 0% to 100% can be used to describe probabilities.

Given, a group of 30 pupils, 13 play the flute only. 1 plays the piano only. 4 play both instruments

From the general formula of probability:

Probability = desired outcome / total outcomes

Total outcomes = 30

In our case desired outcome = is chosen student plays neither instrument

desired outcome =  30 - 13 -1 -4

desired outcome = 12

thus,

Probability = 12/30

probability = 2/5

Therefore, the probability of a randomly selected student playing neither instrument is 2/5.

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For f to be continuous at x=2, b must be equal to 6.  That is, when b=6, there is no break in the function at x=2, and the function is continuous.

To determine the value of b for which f is continuous at x=2, we need to first find the limit of f(x) as x approaches 2 from the left and from the right. From the left, the limit is 4b-4. From the right, it is 2b+2. To make the function continuous at x=2, we need to make sure that the left- and right-hand limits are equal. Setting the two limits equal to each other, 4b-4 = 2b+2, and solving for b yields b=6. Therefore, for f to be continuous at x=2, b must be equal to 6.

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Answer:M<R corresponds to M<T, meaning M<R is 4x+1.

Step-by-step explanation:

The area of the walkway around the garden is 17x+14.

Area is the amount of space occupied by a two-dimensional figure. In other words, it is the quantity that measures the number of unit squares that cover the surface of a closed figure. The standard unit of area is square units which is generally represented as square inches, square feet, etc.

Given that, the area of the garden is 2(5.5x +3.5). The combined area of the garden and the walkway is 3.5(8x + 6).

Area of a walkway = The combined area of the garden and the walkway - The area of the garden

= 3.5(8x+6)-2(5.5x +3.5)

= 28x+21-11x-7

= 17x+14

Therefore, the area of the walkway around the garden is 17x+14.

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(a) P(0≤Z≤2.54) represents the area under the standard normal curve between 0 and 2.54 on the x-axis. This can be found by using a standard normal table or calculator to find the area between 0 and 2.54, which is approximately 0.4966.

(b) P(0≤Z≤2) represents the area under the standard normal curve between 0 and 2 on the x-axis. This can be found by using a standard normal table or calculator to find the area between 0 and 2, which is approximately 0.4772.

(c) P(−2.40≤Z≤0) represents the area under the standard normal curve between −2.40 and 0 on the x-axis. This can be found by using a standard normal table or calculator to find the area between −2.40 and 0, which is approximately 0.3821.

(d) P(−2.40≤Z≤2.40) represents the area under the standard normal curve between −2.40 and 2.40 on the x-axis. This can be found by using a standard normal table or calculator to find the area between −2.40 and 2.40, which is approximately 0.7621.

(e) P(Z≤1.36) represents the area under the standard normal curve to the left of 1.36 on the x-axis. This can be found by using a standard normal table or calculator to find the area to the left of 1.36, which is approximately 0.9155.

(f) P(−1.25≤Z) represents the area under the standard normal curve to the right of −1.25 on the x-axis. This can be found by using a standard normal table or calculator to find the area to the right of −1.25, which is approximately 0.1151.

(g) P(−1.40≤Z≤2.00) represents the area under the standard normal curve between −1.40 and 2.00 on the x-axis. This can be found by using a standard normal table or calculator to find the area between −1.40 and 2.00, which is approximately 0.7170.

(h) P(1.36≤Z≤2.50) represents the area under the standard normal curve between 1.36 and 2.50 on the x-axis. This can be found by using a standard normal table or calculator to find the area between 1.36 and 2.50, which is approximately 0.0951.

(i) P(1.40≤Z) represents the area under the standard normal curve to the right of 1.40 on the x-axis. This can be found by using a standard normal table or calculator to find the area to the right of 1.40, which is approximately 0.0821.

(j) P(|Z|≤2.50) represents the area under the standard normal curve between −2.50 and 2.50 on the x-axis. This can be found by using a standard normal table or calculator to find the area between −2.50 and 2.50, which is approximately 0.5199.

In all cases, a standard normal table or calculator is used to find the area under the standard normal curve. The area represents the probability of the given event.

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The system is inconsistent for k=0.

The system has a unique solution when k=1 and k=-3. To see this, we plug in these values of k into the system and solve the resulting three equations in three unknowns. For k=1, we have x=2, y=1, and z=0. For k=-3, we have x=3, y=2, and z=0.

The system has infinitely many solutions when k=5. In this case, we have x+y-5z=2, x+2y+z=3, and x+y+25z=5, which yields the solution x=2, y=1, and z=0/5. Since any non-zero value of z will also satisfy the system, the system has infinitely many solutions.

The system is inconsistent when k=0. In this case, we have x+y=2, x+2y+z=3, and x+y-z=0, which has no solution. Therefore, the system is inconsistent for k=0.

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Answer:

1) 84

2) 83160

Step-by-step explanation:

Greatest common divisor and Least common multiple

840 = 2³ * 3 * 5 * 7

8316 = 2² * 3³ * 7 * 11

i) The greatest whole number that will divide both 840 and 8316 is the

Greatest common divisor. Find only the common factors.

Greatest common divisor = 2² * 3 * 7

                                          = 84

84 is the greatest whole number that divides both 840 and 8316.

Answer: 84

ii) The smallest non-zero whole number that is divisible by 840 and 8316 is Least common multiple of 840 & 8316.

Least common multiple = 2³ * 3³ * 5 * 7 * 11

                                        = 8 * 27 * 5 * 7 * 11

                                        =83160

Answer:

$5.25

Step-by-step explanation:

Original cost of cheese = $4.20

An increase in price of 25% means the cost increase = 25% of $4.20

25% in decimal = 25/100 = 0.25 (or 1/4)

25% of $4.20 = 0.25 x $4.20 = $1.05

or, if you are more comfortable working with fractions
25% of $4.20 =  1/4 x $4.20 = $1.05

This is only the cost increase.

Add this increase to the original cost to get the new price:

New price = $4.20 + $1.05 =$5.25

The unit rate of change in this situation is 0.25 mile per hour.

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a. Steady state concentration = 10/0.23 = 43.48 kg/m^3

b. Concentration 3 hrs later = 10/0.23 * e^(-3*0.23/1) = 34.99 kg/m^3

a. The steady state concentration of the air pollutant can be found by the equation C = M/k, where C is the concentration, M is the mass loading rate, and k is the reaction rate. In this case, C = 10 kg/s/0.23/hr = 43.48 kg/m^3. b. To calculate the concentration of the pollutant 3 hours later, we can use the equation C = M/k * e^(-kt), where t is the time in hours. In this case, C = 10/0.23 * e^(-3*0.23/1) = 34.99 kg/m^3. This equation takes into account the decrease in wind speed and the decrease in reaction rate due to the decrease in concentration.

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The  basic formula that is used to find out  the area of a triangle is:

Area = (1/2) * base * height.

where "base" is one of the sides of the triangle, and "height" is the perpendicular distance from the vertex opposite the base to the line formed by the base. This formula is based on the fact that the area of a triangle is equal to half of the product of its base and its height. The height is also known as the altitude, and it is the distance from the vertex opposite the base to the line formed by the base. It is important to note that the height must be perpendicular to the base. This formula is valid for any type of triangle, whether it is right-angled or not, and it works well when the base and the height are known. This formula can be used in any triangle, and can be applied to any side as base, not only the bottom side. Another important formula is Heron's formula which is used to find the area of a triangle when the lengths of all three sides are known. Heron's formula is given by:

Area = √(s(s-a)(s-b)(s-c))

where s is said as semi-perimeter of the triangle, and a, b and c are the lengths of the three sides of a given  triangle.

Finally, trigonometric functions like sine, cosine, and tangent can also be used to find the area of a triangle when we know two sides and the angle formed between them. The formula used is:

Area = (1/2) * a * b * sin(C)

where  let us assume a and b are the lengths of two sides of the  given triangle and C is the angle which is  between them.

In conclusion, there are different formulas for calculating the area of a triangle, the most common one is the base times height formula, but depending on the information you have, you can also use Heron's formula or trigonometric functions.

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a. The two numbers whose product is 24 and whose sum is 14 are 6 and 4

b. 14x = 6x + 8x

c. 8x²2² + 6x + 8x + 3

=8x²2² + (6x + 8x) + 3

=8x²2² + 14x + 3

The trinomial 8x²2² + 14x + 3 can be factored by grouping as (8x²2² + 14x) + 3

Yes, 6x -3y = 0 and 2x -4y -3 are parallel. This is because they both have the same slope, -3/2. The slope of a line can be calculated from its equation, by dividing the coefficient of the x term by the coefficient of the y term.

Two lines are parallel if they have the same slope. The slope of a line is represented by the letter "m" in the equation y = mx + b.

In this case, Y=2x + 9 has a slope of 2 and 4x -2y = 3 has a slope of -1/2, these slopes are different, which means that these two lines are not parallel.

On the other hand, 6x -3y = 0 and 2x -4y -3 are parallel because they have the same slope, -3/2. The slope of a line is the amount of change in the y coordinate for each unit of change in the x coordinate.

Yes, 6x -3y = 0 and 2x -4y -3 are parallel. This is because they both have the same slope, -3/2. The slope of a line can be calculated from its equation, by dividing the coefficient of the x term by the coefficient of the y term.

In the case of 6x -3y = 0, the slope is -3/6 = -1/2. In the case of 2x -4y -3, the slope is -4/2 = -2. Since both lines have the same slope, they are parallel. This means that they never intersect and have the same direction.

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The 3 corresponding points that must be on the graph are (1,0),(4,1),(16,2).  for proper calculation please scroll down.

A relation between a collection of inputs and outputs is known as a function. In plain English, a function is an association between inputs in which each input is connected to precisely one output. A domain, codomain, or range exists for every function.

Points on the graph are A(0,1), B(1,4),C(2,16).

Now, these are the points on the graph for the function

y=b^x

Since, we know that the funtion

y=[tex]log_{b}X[/tex]

is the inverse of

y=b^x

Therefore, the points are the reflection of the given point.

So, change the x coordinate with y and y with x to get the points for the function

y=  [tex]log_{b}X[/tex]

The points will become,

(0, 1) ~ (1, 0)

(1, 4) ~ (4, 1)

(2, 16) ~ (16, 2)

Therefore, the 3 corresponding points that must be on the graph are (1, 0), (4, 1), (16, 2).

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The conditional probability that the selected student is male and got grade A is 2/7.

P(A|M) stands for the probability of such a considering that M has happened.

Then, formula for the conditional probability is-

P(A|M) = P(A∩M) / P(M)

P(A∩M)  = Probability that the student is male M and got grade A = 4/25.

P(M) =  Probability that the student is male M = 14/25.

Then,

P(A|M) = 4/25 / 14/25

P(A|M) = 4/14 = 2/7

Thus, the conditional probability that the selected student is male and got grade A is 2/7.

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Step-by-step explanation:

soln on

n(s)=6

n(p)=3

therefore

probability =6/3

The data can be analyzed to determine the distribution of the data.

In order to construct a histogram for the daily sales of IPA beer, in pints, for the first three years of operations of Red Banjo Brewing Company, the lower bin limit is set to 0 and the bin width is set to 50. The data can be analyzed to determine the distribution of the data. To calculate the distribution, the data points are counted and the number of occurrences of each data point is divided by the total number of data points.

Using this formula, the data can be analyzed to determine if the data is symmetric, negatively skewed, positively skewed, or flat. The formula used is:

(Number of Occurrences / Total Number of Data Points) * 100

In the case of the Red Banjo Brewing Company data, the data is found to be positively skewed. This means that there are more data points in the higher range than in the lower range. This suggests that the brewery has seen a steady increase in the sale of IPA beer over the past three years.

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Based on the graph which shows the relationship between time and the number of soda bottles a machine can make, this machine can make 20 soda bottles each minute.

In Geometry, the rate of change of any straight line simply refers to the  slope and it can be calculated by using this mathematical equation (expression);

Rate of change (slope), m = (Change in y-axis, Δy)/(Change in x-axis, Δx)

Rate of change (slope), m = (y₂ - y₁)/(x₂ - x₁)

From the information provided in the graph above, we can logically deduce the following data points located on the line:

Substituting the given data points into the rate of change formula, we have the following;

Rate of change (slope), m = (y₂ - y₁)/(x₂ - x₁)

Rate of change (slope), m = (140 - 100)/(7 - 5)

Rate of change (slope), m = 40/2

Rate of change (slope), m = 20.

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An integral expression, in terms of sine and cosine, that gives the distance the particle travels from t = 1.5 to t = 2.75 is  1.5→2.75 √{(-2sin(πt))²+(4cos(πt))²} dt.

We have been given two functions representing the displacement of the particle,

x(t)=2cos(πt)

y(t)=4sin(πt)

To find the velocity vector with the help of these functions, we have to differentiate them with respect to time

i.e,

dx(t)/dt=d{2cos(πt)}/dt=-2sin(πt)

dy(t)/dt=d{4sin(πt)}/dt=4cos(πt)

and finally, the velocity vector would be written as,

v(t)=-2sin(πt) i +4cos(πt) j

Now to get the distance for the given limits we have to integrate the magnitude of the given velocity

i.e, integrate 1.5→2.75 √{(-2sin(πt))²+(4cos(πt))²} dt

which would approximately result in a value of 3.69 units in length.

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The hypotheses are

We fail to reject the null hypothesis in (b) and (c)

Here, we have

The mean of the 10 strawberry plants differs from 205

So, the hypotheses are

We start by calculating the mean and the sample standard deviation

Using an online calculator, we have

Mean = 223.1

Standard deviation = 40.4

Calculate the t-statistic

t-statistic = (223.1 - 205) / (40.4 / √10) = 1.42

The degrees of freedom for this test are 9 (n - 1).

Using a t-distribution table, the critical value for a one-tailed t-test with 9 degrees of freedom and a 95% level of confidence is 1.833.

Since the calculated t-statistic of 1.42 is less than the critical value of 1.833, we fail to reject the null hypothesis.

Calculate the standard error:

Standard Error = 40.4 / √10 = 12.78

Calculate the z-statistic:

z-statistic = (223.1 - 205) / 12.78 = 1.41

In this case, the z-statistic (1.41) is less than the critical value (1.960) from the standard normal distribution table, so we accept the null hypothesis.

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