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Answer:

[tex]y = 7[/tex]

[tex]x = 7 \sqrt{2} [/tex]

To solve the equation a (a^2 - 16) (a - 2) = 0, we can use the zero product property, which states that if the product of two factors is zero, then at least one of the factors must be zero.

So, we need to find the values of a that make each factor equal to zero.

a = 0 is a solution, since any number times zero is zero.

a^2 - 16 = 0 can be factored as (a - 4)(a + 4) = 0. So, a = 4 or a = -4 are solutions.

a - 2 = 0 gives us a = 2 as a solution.

Therefore, the possible solutions to the equation are a = 0, a = 4, a = -4, and a = 2.

So, the answers to select from are: 0, 6, 4, 16, 1, 2, -4, 8. Among these options, the possible solutions to the equation are 0, 4, -4, and 2.

The required remainder and the quotient is 86 and 4 respectively.

None of the options are correct.

To determine the quotient and remainder by dividing 1534 ÷ 134.

In mathematics, it deals with numbers of operations according to the statements. There are four major arithmetic operators, addition, subtraction, multiplication and division,

The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.

Here,

134 ]  1534  [4

     -448

  --------------

      060

Remainder = 60

Quotient = 11

Thus, the required remainder and quotient is 60 and 11 respectively.

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Therefore , the solution of the given problem of equation comes out to be the equation of the perpendicular bisector of that line segment (7, 3) and (-5, 9) is y = 2x + 4.

Complex algorithms frequently employ variable words to demonstrate coherence between two opposing assertions. Equations are academic expressions that are used to demonstrate the equality of different academic figures. In this case, leveling generates b + 7 rather than another algorithm that could analyze data provided by y + 7, split 12 into two parts, and create y + 7.

Here,

The midpoint formula can be used to determine the midpoint of the line segment whose ends are (7, 3) and (-5, 9):

=> ((7 + (-5))/2, (3 + 9)/2) = (1, 6)

the middle is thus (1, 6). The slope of the line going through (7, 3) and must now be determined. (-5, 9). The slope method is used to:

=>slope = (9 - 3)/(-5 - 7) = -6/12 = -1/2

The equivalent of -1/2 in the negative is 2.

=> y - y1 = m(x - x1)

where (x1, y1) is a location on the line and m denotes the slope. When we replace m = 2 with (1, x1, y1) = (1, 6), we obtain:

=> y - 6 = 2(x - 1)

By enlarging and condensing, we obtain:

=> y - 6 = 2x - 2

=> y = 2x + 4

the line segment whose ends are and the equation of the perpendicular bisector of that line segment (7, 3) and (-5, 9) is y = 2x + 4.

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Answer:

To calculate the monthly payment for the TV after the discount, sales tax, and financing, we need to follow these steps:

Step 1: Calculate the discount

Discount = 25% of $987.48

Discount = 0.25 x $987.48

Discount = $246.87

Step 2: Calculate the discounted price of the TV

Price after discount = $987.48 - $246.87

Price after discount = $740.61

Step 3: Calculate the sales tax

Sales Tax = 6.5% of $740.61

Sales Tax = 0.065 x $740.61

Sales Tax = $48.15

Step 4: Calculate the total cost of the TV after discount and sales tax

Total cost = Price after discount + Sales Tax

Total cost = $740.61 + $48.15

Total cost = $788.76

Step 5: Calculate the monthly payment on the 2-year loan with 15.6% annual interest rate using the following formula:

Monthly Payment = (P x r) / (1 - (1 + r)^(-n))

where P is the principal (total cost of the TV), r is the monthly interest rate (15.6% / 12), and n is the number of months (24).

Monthly Interest Rate = 15.6% / 12

Monthly Interest Rate = 0.013

Monthly Payment = ($788.76 x 0.013) / (1 - (1 + 0.013)^(-24))

Monthly Payment = $36.89

Therefore, the monthly payment on the 2-year loan for the TV is $36.89.

Step-by-step explanation:

Substitute a=7, b=5, and c=3 into the expression ab+c:

ab+c = (7)(5) + 3

ab+c = 35 + 3

ab+c = 38

Therefore, ab+c is equal to 38 when a=7, b=5, and c=3.

Answer: It would be 3:13

Step-by-step explanation:

Pls brainliest if this helped you! :D

Answer:

3:13

Step-by-step explanation:

First we have to find the GCF (greatest common factor of both numbers) which is 20

So 60 divided by 20 is 3 and 260 divided by 20 is 13

Your final answer should by 3:13 because you can't divide any further using those numbers or any other number

Hope this helps!

Also for your information to make things easier --> keep in mind that whenever it says "ratio" means "to divide" !!!

The counting process was an essential component in evaluating the results and drawing conclusions based on the data gathered.

Hi! Bob and Aniyah's apples were counted by a designated individual, likely a teacher, supervisor, or event organizer, who was responsible for ensuring the accuracy and fairness of the counting process.

This person utilized their organizational skills and attention to detail to effectively tally the number of apples collected by both Bob and Aniyah.

By doing so, they were able to determine the final count, which was then used to either compare the performance of the two individuals or contribute to a larger total for a specific event or purpose.

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The measures for the triangle are given as follows:

The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

The theorem is expressed as follows:

c² = a² + b².

In which:

The base is composed by two lengths of 6 and 25 - 6 = 19, hence the value of x is obtained applying the Geometric Mean Theorem as follows:

x² = 6 x 19

x = sqrt(6 x 19)

x = 10.7.

The value of y is then the hypotenuse of a right triangles of sides 10.7 and 6, hence:

y² = 10.7² + 6²

y = sqrt(10.7² + 6²)

y = 12.3.

The value of z is then the hypotenuse of a right triangles of sides 10.7 and 19, hence:

z² = 10.7² + 19²

z = sqrt(10.7² + 19²)

z = 21.8.

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We can factor the left side of the equation and use the zero product property to find possible values of a. We get a=-3 and a=2. By substituting these values back into the original equation, we can find the corresponding values of b which are -18 and 12, respectively.

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols to solve equations and understand relationships between variables. It involves using letters and symbols to represent numbers and quantities, and manipulating equations to solve for unknown variables. Algebra is used in various fields of study, including science, engineering, economics, and finance.

To solve for the possible values of a and b given the equation:

We can use algebraic manipulation to rewrite the equation in terms of one variable.

Starting with:

we can simplify the left side of the equation by factoring out a common factor of (a + 3):

Now, we can use the zero product property, which states that if the product of two factors is equal to zero, then at least one of the factors must be zero.

Therefore, we have two possibilities:

(a + 3) = 0

If a + 3 = 0, then a = -3. Substituting this value of a back into the original equation, we get:

(2a - 3) = 0

If 2a - 3 = 0, then a = 3/2. Substituting this value of a back into the original equation, we get:

Now we have found the possible values of a. To find the corresponding values of b, we can substitute each value of a into either of the original equations and solve for b. Let's use the first equation:

When a = -3, we have:

Simplifying, we get:

-3b = -54

Dividing both sides by -3, we get:

b = 18

Therefore, when a = -3, b = 18.

When a = 2, we have:

Simplifying, we get:

2b = 12

Dividing both sides by 2, we get:

b = 6

Therefore, when a = 2, b = 6.

Thus, the possible values of a are -3 and 2, and the corresponding values of b are -18 and 12, respectively.

Therefore,We can factor the left side of the equation and use the zero product property to find possible values of a. We get a=-3 and a=2. By substituting these values back into the original equation, we can find the corresponding values of b which are -18 and 12, respectively

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The selling price of the oranges was 472.5 dollars.

The profit is equal to the difference between the selling price and the original cost price.

We have been given that a basket containing 150 oranges costs 450 find the selling price of an orange if it is to be sold at a profit of 5%

Let's start with the profit.

Let's convert 5% to decimal form to find out.

[tex]\implies 5\% = \dfrac{5}{100} = 0.05[/tex]

Then double that by the starting price of 450.

[tex]450\times 0.05 = 22.5[/tex]

As a result, the profit is $22.5.

Now, add the profit to the beginning cost to get the selling price.

[tex]450 + 22.5 = 472.5[/tex]

Therefore, the selling price of the oranges was 472.5 dollars.

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Answer:

what number is right before "of the students ordered chicken nuggets and the rest of the students are eating nachos." you cant answer the question otherwise.

Step-by-step explanation:

Answer: Actually, the correct answer is:

−0.1 + 1.7 = 1.6

So, y = 1.6.

Step-by-step explanation:

The expected value of the car representing the delay of the car at one of the light as per distribution of time is given by option d. 2/3.

For the probability density function of X.

Divide the interval from 0 to 3 minutes into three sub-intervals,

The first minute, during which the car will pass the intersection if the light is green.

The second minute, during which the car will be delayed by one minute if the light is red.

The third minute, during which the car will be delayed by two minutes if the light is red.

The probability of the car arriving during the first sub-interval is 1/3.

If the light is green, the car will pass immediately with no delay.

If the light is red, the car will be delayed by one minute.

So the probability density function of the delay during this sub-interval is,

f(x) ={  1/3, if 0 ≤ x ≤ 1

         0,     otherwise  }

Probability of the car arriving during the second sub-interval is also 1/3.

If the light is red, the car will be delayed by one minute.

If the light is green, the car will be delayed by two minutes.

So the probability density function of the delay during this sub-interval is,

f(x) = { 1/6,      if 1 ≤ x ≤ 2

         1/3,       if 2 ≤ x ≤ 3

         0,          otherwise }

Probability of the car arriving during the third sub-interval is also 1/3.

If the light is red, the car will be delayed by two minutes.

If the light is green, the car will be delayed by three minutes.

But since the car cannot wait for more than three minutes, the delay during this sub-interval is bounded by 2 minutes.

So the probability density function of the delay during this sub-interval is,

f(x) = {  1/6,     if 2 ≤ x ≤ 3

             0,      otherwise  }

Now,

calculate the expected value of X by integrating the product of the delay and the probability density function over the entire range of possible delays

E(X) = [tex]\int_{0}^{3}[/tex] x f(x) dx

= [tex]\int_{0}^{1}[/tex]0 dx + [tex]\int_{1}^{2}[/tex] x (1/6) dx + [tex]\int_{2}^{3}[/tex] x(1/6) dx

= (1/6)(1/2)  [(2^2 - 1^2) + (3^2 - 2^2)]

= (1/12) [ 3 + 5 ]

= 8/12

= 2/3

Therefore, the expected value of the delay of the car at the light is equal to option d. 2/3.

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Answer: y=3x

Step-by-step explanation: To find the equation of the line m, which is the image of line t after a dilation with a scale factor of 1/2 centered at the origin, we need to follow these steps:

Rewrite the equation of line t in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.

3X - Y = 6

-Y = -3X + 6

Y = 3X - 6

So the slope of line t is 3.

Apply the dilation with a scale factor of 1/2 centered at the origin. This means that every point on line t is multiplied by 1/2 in both the x and y directions. The origin (0,0) remains fixed.

The new coordinates of any point on line t after dilation are given by (1/2x, 1/2y).

Use the new coordinates to find the equation of line m. To do this, we need to find the slope of line m and a point on it.

The slope of line m is the same as the slope of line t, which is 3.

To find a point on line m, we can use the fact that the origin remains fixed after dilation. So the point (0,0) is on line m.

Using the slope-intercept form, y = mx + b, we can write the equation of line m as:

y = 3x + b

To find the value of b, we can use the point (0,0):

0 = 3(0) + b

b = 0

Therefore, the equation of line m is y = 3x.

The Equation of line m is y= 3x - 3.

A scale factor is a number that can be used to adjust the size of any geometrical figure or object in relation to its original size. It's used to draw the enlarged or reduced shape of any given figure, as well as to calculate the missing length, area, or volume of an enlarged or reduced figure. It should be noted that the scale factor only affects the figure's size and not its shape.

We have, Equation of line t

3x - y = 6

Take x= 0 then y = -6 and y= 0 then x = 2.

Now, dilate the coordinates by factor 1/2 as

(0, -3) and (1, 0).

So, the Equation of line m is

(x - 0) / (1- 0) = (y - (-3)) / (0 - (-3))

x/1 = y + 3 / 3

3x = y+ 3

y= 3x - 3

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Thus, the number of buses needed to carry the 2177 tourists is found as 61.

A number can be divided into an equal amount of component parts.

People may display a variety of signs to denote division. The backslash / is additionally employed, though the character is more typical. Numerous numbers may occasionally be written with a line separating them. It is also known as a fraction.

A division calculation has a name for each component. The dividend, the divisor, and indeed the quotient are the three key terms.

Given data:

Number of tourists carried by 1 bus = 36.

Total number of tourists = 2177.

Number of buses = (Total number of tourists)/(Number of tourists carried by 1 bus)

Number of buses = 2177/36

Number of buses = 60.47

Number of buses =  61

Thus, the number of buses needed to carry the 2177 tourists is found as 61.

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The point that appears to be closest to the coordinates (29, 37.7) on the scatter plot is (29,37).

A scatter plot is a type of chart that displays values for two variables for a set of data. The data points in a scatter plot are represented by dots or circles, and their position on the chart represents the values of the two variables. One variable is usually plotted on the horizontal axis (x-axis) and the other variable is plotted on the vertical axis (y-axis). By analyzing the pattern of the scatter plot, you can see whether there is a relationship between the two variables, and what kind of relationship it is (positive, negative, or no relationship). Scatter plots are useful for identifying patterns, trends, and relationships in data, and are commonly used in scientific research, engineering, economics, and many other fields.

Here,

To find the foot length closest to 29, we need to look for the point on the scatter plot that has an x-value closest to 29.

Using the equation of the line provided (y=0.3x+29), we can see that when x=29, y=0.3(29)+29=37.7. Therefore, we are looking for the point on the scatter plot with an x-value close to 29 and a y-value close to 37.7.

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Complete question:

Here is a scatter plot that shows the lengths and widths of `20` left feet together with the graph of a model of the relationship between foot length and width. Circle the point that represents the foot length closest to `29`.

Answer:

Step-by-step explanation:

34,000,000,000=4,005,000(1+0.041)^x

(1.041)^x=(34,000,000,000)/4,005,000

(1.041)^x=34,000,000/4,005

xlog(1.041)=log34,000,000-log 4005

=7.15348-3.60260

=3.55088

x=3.55088/log(1.041)=3.55088/0.01745

\approx 203.5~years

Answer:

Step-by-step explanation:

[tex]FV = (PMT * \frac{(1+i)^{n} -1}{i} )* (1 + i)[/tex]

FV = Future Value of the annuity

PMT = Amount of each annuity payment

i = Interest rate per period

n = Number of periods for which annuity will last

FV = ?

PMT = $4,500

i = 5.2% = 5.2/100 = 0.052

n = 3

[tex]FV = (4500 * \frac{(1+0.052)^{3} -1}{0.052} )* (1 + 0.052)[/tex]

[tex]FV = 14,953.30[/tex]

You will have $14,953.30 in 3 years.

Answer: b I believe.

Step-by-step explanation:

4,500 times 5.2% = 4,754 add that to the amount of money you have already. giving you b.

So the price that would maximize revenue is $2.75.

An equation is a mathematical statement that asserts the equality of two expressions. It typically contains one or more variables, which represent unknown values that can be solved for using algebraic methods. An equation can be represented symbolically using mathematical notation, such as using letters, numbers, and mathematical symbols like +, -, x, /, and =. Equations can be used to model and solve a wide variety of problems in mathematics, science, engineering, and other fields.

Here,

We can start by finding the linear equation that relates price and monthly sales. Let's use the point-slope form of a linear equation:

y - y1 = m(x - x1)

where y is the monthly sales, x is the price, m is the slope, and (x1, y1) is a point on the line.

Using the two points we have:

(10, 19000) and (7, 25000)

We can find the slope:

m = (y2 - y1) / (x2 - x1)

m = (25000 - 19000) / (7 - 10)

m = 2000

Now we have:

y - 19000 = 2000(x - 10)

Simplifying:

y = 2000x - 11000

This equation represents the relationship between price and monthly sales.

To maximize revenue, we need to find the price that gives the highest value for the product of price and monthly sales. In other words, we need to find the maximum value of the function:

R(x) = x * y = x(2000x - 11000)

Expanding and simplifying:

R(x) = 2000x² - 11000x

To find the maximum, we need to take the derivative and set it equal to zero:

R'(x) = 4000x - 11000 = 0

Solving for x:

4000x = 11000

x = 2.75

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Answer: I believe it is option (A): 4.5

Step-by-step explanation: If you look closely and examine DE and CE you will see they are the same length, and we know they are 5. But if you compare the lines from a diagonal perspective, BD is not equal to 5 and has to be a bit smaller. The closest option below 5 is therefore 4.5

Hope this helps! Please mark brainliest!

Answer : 52.

Explanation : For Triangles ( 3 × 4 ) ÷ 2     Rectangle 10 × 4

The area of this quadrilateral is 52 square feet.

First, I noticed that the rectangle had sides which were 10 feet long and 4 feet tall. (10 × 4)

Second, I noticed that the triangles were 3 feet long and 4 feet tall. (3 × 4) Since this was a triangle, I then divided by 2. (3 × 4) ÷ 2 I then saw there were 2 triangles so I multiplied it by 2. [(3 × 4) ÷ 2] × 2

Lastly, the equation I was left with was (10 × 4) + {[(3 × 4) ÷ 2] × 2} = 5

—————

Please remember to revise this and make it in your own words if you want! I hope this helps you. -Doodle

—————

Bus 14 is therefore the more reliable bus as a result of its lower IQR.

The dot plot, which depicts the frequency with which each observation appears in the data set, is the first thing we take into account.

The difference between the data set's third and first quartiles is represented by the interquartile range, which is the next factor we take into account.

Because it ignores outliers, the interquartile range is a better indicator of a spread than a data set's range.

We have the following for classes of 15 students:

Since the first seven students make up the first half, the first quartile is represented by the fourth dot, which represents the first half's median.

The final seven students make up the second half, thus the eleventh dot—the first half's median—is in the first quartile.

So, the following are the Bus 14 quartiles:

Q1 = 12.

Q3 = 18.

The IQR is therefore of:

IQR = Q3 - Q1 = 18 - 12 = 6.

Hence, the IQR is:

Q1 = 9.

Q3 = 16.

The IQR is therefore of:

IQR = Q3 - Q1 = 16 - 9 = 7.

Therefore, bus 14 is therefore the more reliable bus as a result of its lower IQR.

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Correct question:

The line plots represent data collected on the travel times to school from two groups of 15 students. A horizontal line starting at 0, with tick, marks every two units up to 28. The line is labeled Minutes Traveled. There is one dot above 10,16,20, and 28. There are two dots above 8 and 14. There are three dots above 18. There are four dots above 12. The graph is titled Bus 14 Travel Times. A horizontal line starting at 0, with tick, marks every two units up to 28. The line is labeled Minutes Traveled. There is one dot above 8, 9,18, 20, and 22. There are two dots above 6, 10, 12,14, and 16. The graph is titled Bus 18 Travel Times. Compare the data and use the correct measure of variability to determine which bus is the most consistent. Explain your answer. Bus 14, with an IQR of 6 Bus 18, with an IQR of 7 Bus 14, with a range of 6 Bus 18, with a range of 7

Sο, if οur estimatiοns are reasοnably accurate, the likelihοοd that a chοsen student is a wοman οr a business majοr is rοughly 0.6.

It describes the likelihοοd οf a specific οccurrence οr grοup οf οutcοmes in a given envirοnment and is a key idea in mathematics and statistics. Prοbability is typically stated as a number between 0 and 1, with 1 denοting a certain event and 0 denοting an impοssibility. By dividing the number οf faοurable οutcοmesv by the tοtal number οf pοssible οutcοmes, οne can determine the prοbability οf an οccurring.

We must sum the prοbabilities οf twο events—the prοbability that the student is a wοman and the prοbability that the student is a business majοr—in οrder tο get the likelihοοd that a certain student is a wοman οr a business majοr.

P(A and B) equals P(A)*P (B)

Using this presumptiοn, we may calculate the likelihοοd that a chοsen student will majοr in business and be a wοman as fοllοws:

P (wοman) = 0.2 * P (business majοr) = 0.1 (10%)

After that, we can enter this value in the previοus equatiοn:

P (wοmen majοring in business) = 0.5 + 0.2 - 0.1 = 0.6 (οr 60%)

Sο, if οur estimatiοns are reasοnably accurate, the likelihοοd that a chοsen student is a wοman οr a business majοr is rοughly 0.6.

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Complete Question:

A college has an undergraduate enrollment of 3500. Of these, 860 are business majors, 1800 are women and 425 are women business majors.

a. Are the events "selecting a woman student" and "selecting a business major" mutually exclusive? Explain.

b. If a college newspaper conducts a poll and selects students at random to answer a survey, find the probability that a selected student is a woman or a business major.

The area of ΔQRS is 26.10 square units.

What is triangle?

A triangle is a closed, two-dimensional geometric shape with three straight sides and three angles.

To find the area of [tex]$\triangle QRS$[/tex], we can use the formula:

[tex]$Area = \frac{1}{2} \times base \times height$[/tex]

where the base and height are the length of two sides of the triangle that are perpendicular to each other. We can find these sides using trigonometry.

First, we need to find the length of side [tex]$QR$[/tex]. We can use the Law of Cosines:

[tex]$QR^2 = QS^2 + RS^2 - 2(QS)(RS)\cos(R)$[/tex]

where [tex]$R$[/tex] is the angle at vertex [tex]$R$[/tex]. Substituting the given values, we get:

[tex]$QR^2 = 9^2 + 5^2 - 2(9)(5)\cos(57^\circ)$[/tex]

[tex]$QR \approx 8.02$[/tex]

Next, we need to find the height of the triangle, which is the perpendicular distance from vertex [tex]$R$[/tex] to side [tex]$QS$[/tex]. We can use the sine function:

[tex]$\sin(R) = \frac{opposite}{hypotenuse}$[/tex]

[tex]$\sin(57^\circ) = \frac{height}{8.02}$[/tex]

[tex]$height \approx 6.51$[/tex]

Now we can find the area of the triangle:

[tex]$Area = \frac{1}{2} \times QR \times height$[/tex]

[tex]$Area = \frac{1}{2} \times 8.02 \times 6.51$[/tex]

[tex]$Area \approx 26.10$[/tex] square units

Therefore, the area of [tex]$\triangle QRS$[/tex] is approximately [tex]$26.10$[/tex] square units.

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The expression that corresponds to the provided sentence is 2sinx cos²x.

As an illustration, the expression x + y is one where both x and y have words with an addition function in between. There are two kinds of expressions in mathematics: numerical expressions, which only comprise integers, and algebraic expressions, which also include variables.

In mathematics, an expression is a combination of one or more numbers, variables, and mathematical operations (such as addition, subtraction, multiplication, division, exponents, and roots) that can be evaluated or simplified to produce a single value.

For example, 2 + 3x is an expression that contains the variables x and the numbers 2 and 3, as well as the operations of addition and multiplication.

Another example is [tex](4a^2 - 3b)/c[/tex], which is an expression that contains the variables a, b, and c, as well as the numbers 4 and 3, and the operations of addition, subtraction, multiplication, and division.

sin2x cos x is equivalent to (2sinx cosx) cosx,

which simplifies to 2sinx cos²x.

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question-

Which of the following expressions is equal to sin(2x) 2(1 - cos^2(x))

Answer:

4.  4/13

5.  7/13

6.  4/13

Step-by-step explanation:

A standard deck has 52 cards, 4 suits, each suit has cards Ace through 10 and the face cards, Jack, Queen, King.

If the idea of suits confuses you, think about Uno where the cards are blue, yellow, green, and red. Only here the suits are Diamonds, Clubs, Hearts, and Spades

To calculate compound probability use the following formula:

P(A ∪ B) = P(A) + P(B) - P(A | B)

This says that the probability of A or B occuring is the probability of A, plus the probability of B, minus the probability of them both occuring at the same time.

4)  Randomly selecting a diamond or a seven.

There are 13 diamonds in a deck and 4 sevens, one from each suit. The joint event for this probabaility is randomly drawing a 7 of diamonds. There is only one of these in the deck.

P(Diamond ∪ 7) = P( Diamond ) + P( 7 ) - P( Diamond | 7)

                         = 13/52 + 4/52 - 1/52

                         = 16/52 = 4/13

5)  Randomly selecting a red card or a queen.

There are 26 red cards in the deck, from diamonds and hearts suits, 13 from each. There are 4 queens in the deck, one from each suit. The joint event is a queen of a diamonds or hearts, which there are 2 of in the deck.

P( Red ∪ Queen ) = P( Red ) + P( Queen ) - P( Red | Queen)

                             = 26/52 + 4/52 - 2/52

                             = 28/52 = 7/13

6) Randomly selecting a three or a face card.

There are 4 threes in a deck of cards, one from each suit. There are 12 face cards in a deck (Jack, Queen, and King with three from each suit). There is no joint event because you cannot draw a three and a face card. So this time we subtract 0 because the events are mutually exclusive.

P( 3 | Face ) = P( 3 ) + P( Face ) - P( 3 | Face )

                   = 4/52 + 12/52 - 0/52

                   = 16/52 = 4/13

The average rate of change from year 1 to year 10 is 8.3 pounds per acre per year based on given function.

We must compute the change in crop yield (in pounds per acre) during that time period and divide by the number of years to determine the average rate of change from year 1 to year 10:

Average change rate is equal to (f(10) - f(1)) / (10 - 1)

where f(x): function that represents the crop yield over an x-year period in pounds per acre.

When we enter the numbers 10 and 1 as substitutes in the function, we obtain:

f(10) = [tex]-10^2[/tex] + 20(10) + 50 = 150

f(1) =[tex]-1^2[/tex] + 20(1) + 50 = 69

Thus, from year one to year ten, the average rate of change is:

(150 - 69) / (10 - 1), or 8.3 pounds per acre per year, is the average rate of change.

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Solving equations and inequalities are essential skills in various real-world applications, from engineering and physics to business and finance.

These skills enable us to find solutions to problems and make informed decisions. For instance, engineers use equations to design buildings, bridges, and machines, while scientists use equations to model and predict the behavior of physical systems. In business and finance, equations and inequalities are used to solve problems related to budgets, investments, and financial planning.

They help individuals and organizations make informed decisions and manage resources efficiently. For example, a company can use equations to determine the optimal production level that maximizes profits or to evaluate the feasibility of a new project.

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